## 3 Curvature Interference Limit Function of Meshing
Finally, these results are employed to perform curvature interference analysis. The flowchart of the proposed tooth surface contact analysis method based on PINNs is illustrated in Figure 6. The detailed derivation of the loss function will be presented in the subsequent subsection. The calculation of curvature interference limit lines can be reduced to determining key points on the curvature interference limit lines. Based on this, a loss function can be established. To determine the geometric boundary conditions of the tooth surface, the worm and face gear tooth surfaces are projected onto their axial cross-sections, as illustrated in Figure 7a,b. As illustrated in Figure 7, within the axial section coordinate system \(\left(O_{1};\sqrt{x_{2}^{2}+y_{2}^{2}},y_{\mathrm{o1}}^{*}\right)\) of the face worm gear, the vertical coordinates \(y_{\mathrm{o1}}^{*}=Y_{\mathrm{P}}+c\) and \(y_{\mathrm{o1}}^{*}=Y_{\mathrm{P}}-h_{\mathrm{u2}}\) of the dedendum AD and addendum BC define the upper and lower boundaries of the relevant region. Within this range, the vertical coordinate of the key point \(\mathrm{P}_{j}\) can be expressed as \(y_{\mathrm{o1}}^{*}=L\). Subsequently, by combining \(y_{\mathrm{o1}}^{*}=L\) with \(\phi_{\mathrm{d}}(u,\theta)=0\) in Equation (6), \(\phi(u,\theta,\varphi,\varphi_{1})=0\) in Equation (13), and \(\psi(u,\theta,\varphi,\varphi_{1})=0\) in Equation (18), a system of four nonlinear equations is derived for solving the key point \(\mathrm{P}_{j}\) on the curvature interference limit line. This system is expressed as:  
\[\left\{ \begin{array}{l}y_{\mathrm{o1}}^{*}=L\\ \phi_{\mathrm{d}}(u,\theta)=0\\ \phi(u,\theta,\varphi,\varphi_{1})=0\\ \psi(u,\theta,\varphi,\varphi_{1})=0\\ \end{array} \right.. \tag{19}\]  
Figure 6: Flowchart of curvature interference characteristic analysis using PINN based on curvature interference theory. To solve Equation (19), the elimination method is initially employed to eliminate variables of Equation (19). Therefore, from Equation (7), when \(A_{\text{d}}\
eq 0\), \(\phi_{\text{d}}(u,\theta)=0\) can be rewritten as  
\[u=-\frac{B_{\text{d}}(\theta)}{A_{\text{d}}(\theta)}. \tag{20}\]  
By simultaneously considering \(\phi=0\) and \(y_{\text{o1}}^{*}=L\) in Equation (19), it is possible to derive it as:  
\[\left\{ \begin{array}{l}A\sin(\varphi_{1}-\varphi)+B\cos(\varphi_{1}-\varphi )=-C\\ x_{\text{o1}}\sin(\varphi_{1}-\varphi)+(y_{\text{o1}}+a_{\text{d}})\cos(\varphi _{1}-\varphi)=L\end{array} \right.. \tag{21}\]  
Consequently, the trigonometric functions \(\sin(\varphi_{1}-\varphi)\) and \(\cos(\varphi_{1}-\varphi)\) can be derived using Equation (21) as  
\[\sin(\varphi_{1}-\varphi)=\frac{\left| \begin{array}{cc}L&y_{\text{o1}}+a_{ \text{31}}\\ -C&B\\ \hline x_{\text{o1}}&y_{\text{o1}}+a_{\text{31}}\\ A&B\end{array} \right|}{\left| \begin{array}{cc}L&y_{\text{o1}}+a_{\text{d}} \\ \hline x_{\text{o1}}B-(y_{\text{o1}}+a_{\text{d}})A\end{array} \right|}=\frac{LB +(y_{\text{o1}}+a_{\text{d}})C}{x_{\text{o1}}B-(y_{\text{o1}}+a_{\text{d}})A },\ \cos(\varphi_{1}-\varphi)=\frac{\left| \begin{array}{cc}x_{\text{o1}}&L\\ A&-C\\ \hline x_{\text{o1}}&y_{\text{o1}}+a_{\text{d}}\\ A&B\end{array} \right|}{\left| \begin{array}{cc}x_{\text{o1}}&L\\ A&-C\\ \hline x_{\text{o1}}&y_{\text{o1}}+a_{\text{d}}\\ A&B\end{array} \right|}=\frac{-x_{\text{o1}}C-LA}{x_{\text{o1}}B-(y_{\text{o1}}+ a_{\text{d}})A}.