# Research on Effective Design Methods of Core Beam of Full Bridge Aeroelastic Model
## 3 Numerical Solution of Torsion Constant
### Generalized Difference Method  
In 1978, [PERSON] et al. [35] merged the integral interpolation method into the generalized Galerkin method and thereby extended the irregular grid difference method to the generalized difference. This method not only maintains the simplicity of the difference method, but also has the accuracy of the finite element.  
The above method adopts a triangular meshing scheme, as shown in Figure 2. The entire cross-section region \(\Omega\) is triangularly meshed and the internal angle of each triangle is less than \(90^{\circ}\). Taking \(P_{0}\) in the subregion \(G_{0}\) as an example, outer nodes around \(P_{0}\), namely \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), \(P_{5}\), and \(P_{6}\), are selected, and these points are arranged counterclockwise. Meanwhile, six triangles, \(P_{0}P_{1}P_{2}\), \(P_{0}P_{2}P_{3}\), \(P_{0}P_{3}P_{4}\), \(P_{0}P_{4}P_{5}\), \(P_{0}P_{5}P_{6}\) and \(P_{0}P_{6}P_{1}\), are defined by these seven points with point \(P_{0}\) being their common vertex. The outer centers of these six triangles, \(q_{1}\), \(q_{2}\), \(q_{3}\), \(q_{4}\), \(q_{5}\) and \(q_{6}\), and the adjacent-side midpoints of the six triangles, \(m_{1}\), \(m_{2}\), \(m_{3}\), \(m_{4}\), \(m_{5}\) and \(m_{6}\), define the shaded part \(G_{0}\), which also is a dual unit.  
For any node inside the region \(\Omega\), a difference format is created. Both sides of Equation (1) are integrated over the subdomain \(G_{0}\):  
\[\iint\limits_{G_{0}}\
abla^{2}\phi dxdy=\iint\limits_{G_{0}}-2 dxdy, \tag{4}\]  
Using the Green formula, Equation (4) is rewritten as  
\[-\int_{\partial G_{0}}\frac{\partial\phi}{\partial n}ds=\iint\limits_{G_{0}}-2 dxdy \tag{5}\]  
where, \(\partial G_{0}\) is the boundary of the subdomain \(G_{0}\), and n is the unit normal vector of \(\partial G_{0}\).  
Therefore, the difference format for each point in the subdomain \(G_{0}\) is obtained as  
\[\int_{\partial G_{0}}\frac{\partial\phi}{\partial n}ds=\sum\limits_{i=1}^{6} \frac{\overline{q_{i}q_{i+1}}}{\overline{P_{0}P_{i+1}}}\cdot[\phi(P_{i+1})- \phi(P_{0})]+m(G_{0})R_{G_{0}}(\phi),(q_{7}=q_{1}), \tag{6}\]  
where, m(\(G_{0}\)) is the area of \(G_{0}\), \(RG_{0}(\phi)\) is the truncation error.  
Substituting Equation (6) into Equation (3) and rounding \(RG_{0}(\phi)\) off, the difference equation of point \(P_{0}\) is presented as Equation (7):  
\[\begin{array}{l}-\sum\limits_{i=1}^{6}\frac{\overline{q_{i}q_{i+1}}}{\overline {P_{0}P_{i+1}}}[\phi(P_{i+1})-\phi(P_{0})]=\iint\limits_{G_{0}}-2 dxdy=m(G_{0} )\cdot\varphi_{0}\\ \varphi_{0}=\frac{1}{m(G_{0})}\iint\limits_{G_{0}}-2 dxdy\end{array}, \tag{7}\]  
For an arbitrary point on the boundary \(\Gamma\), the stress function value of \(P_{0}\), is obtained as  
\[\phi(P_{0})=0, \tag{8}\]  
The difference equations of all points are formed as a closed system of linear algebraic equations, and its coefficient matrix is symmetric. After solving the equations, the stress results in the region \(\Omega\) are obtained as shown in Figure 3. According to Equation (3), the double volume of the stress results is the torsion constant of the U-shaped cross-section.  
Figure 2: Geometric structure of triangle difference scheme.