# Research on Effective Design Methods of Core Beam of Full Bridge Aeroelastic Model
## 3 Numerical Solution of Torsion Constant
### Five-Point Difference Method  
In the five-point difference method, a partial derivative is replaced by the difference quotient of several adjacent numerical points [36], which applies well to a rectangular section. First, a rectangular section is meshed into grids as shown in Figure 5. The edge along the \(x\)-axis is divided into \(N\) equal segments and the one along the \(y\)-axis is divided into \(M\) equal segments. Let \(x_{i}=a+ih_{1}\), \(y_{j}=c+ih_{2}\), where \(0\leq i\leq N\), and \(0\leq j\leq M\). In addition, \(\phi_{\bar{y}}=\phi(x_{i},y_{j})\) is defined and the Poisson problem at the inner node is defined as  
\[\frac{\partial^{2}\phi\big{(}x_{i},y_{j}\big{)}}{\partial x^{2}}+\frac{ \partial^{2}\phi\big{(}x_{i},y_{j}\big{)}}{\partial y^{2}}=const\big{(}x_{i}, y_{j}\big{)},1\leq i\leq N-1,0\leq j\leq M-1, \tag{16}\]  
where, \(\text{const}(x_{i},y_{j})\) is a constant value equal to \(-2\) in Equation (13). The two terms in the left side of Equation (16) are expanded according to the [PERSON] formula, as  
\[\frac{\partial^{2}\phi\big{(}x_{i},y_{j}\big{)}}{\partial x^{2}}=\frac{1}{h_{1 }^{2}}\big{[}\phi\big{(}x_{i-1},y_{j}\big{)}-2\phi\big{(}x_{i},y_{j}\big{)}+ \phi\big{(}x_{i+1},y_{j}\big{)}\big{]}-\frac{h_{1}^{2}}{12}\frac{\partial^{4} \phi\big{(}\xi_{ij},y_{j}\big{)}}{\partial x^{4}},x_{i-1}<\xi_{ij}<x_{i+1} \tag{17}\]  
\[\frac{\partial^{2}\phi\big{(}x_{i},y_{j}\big{)}}{\partial y^{2}}=\frac{1}{h_{2 }^{2}}\big{[}\phi\big{(}x_{i},y_{j-1}\big{)}-2\phi\big{(}x_{i},y_{j}\big{)}+ \phi\big{(}x_{i},y_{j+1}\big{)}\big{]}-\frac{h_{2}^{2}}{12}\frac{\partial^{4} \phi\big{(}x_{i},\eta_{ij}\big{)}}{\partial y^{4}},y_{i-1}<\eta_{ij}<y_{i+1} \tag{18}\]  
Substituting Equation (17) and (18) into Equation (16) and ignoring the second order quantity \(O({h_{1}}^{2}+{h_{2}}^{2})\), one obtains  
\[\frac{1}{h_{2}^{2}}\phi_{i,j-1}\frac{1}{h_{1}^{2}}\phi_{i-1,j}-2\left(\frac{1 }{h_{1}^{2}}+\frac{1}{h_{2}^{2}}\right)\phi_{i,j}+\frac{1}{h_{1}^{2}}\phi_{i+1,j}+\frac{1}{h_{2}^{2}}\phi_{i,j+1}=const\big{(}x_{i},y_{j}\big{)},1\leq i\leq N -1,0\leq j\leq M-1, \tag{19}\]  
Figure 4: Finite element stress function. Figure 5: A rectangular cross-section. The stress function vector \(\phi_{j}\) is defined as  
\[\mathbf{\phi}_{j}=\big{(}\phi_{1j},\quad\phi_{2j},\dots,\quad\phi_{N-1,j}\big{)}^{T}, 0\leq j\leq M, \tag{20}\]  
and Equation (19) is rewritten as  
\[\mathbf{D}\mathbf{\phi}_{j-1}+\mathbf{C}\mathbf{\phi}_{j}+\mathbf{D}\mathbf{\phi}_{j+1}=\mathbf{const}_{j}, 1\leq j\leq M-1, \tag{21}\]  
\[\mathbf{C}= \begin{pmatrix}2\left(\frac{1}{h_{1}^{2}}+\frac{1}{h_{2}^{2}}\right)&- \frac{1}{h_{1}^{2}}\\ -\frac{1}{h_{1}^{2}}&2\left(\frac{1}{h_{1}^{2}}+\frac{1}{h_{2}^{2}}\right)&- \frac{1}{h_{1}^{2}}\\ \ddots&\ddots&\ddots\\ &-\frac{1}{h_{1}^{2}}&2\left(\frac{1}{h_{1}^{2}}+\frac{1}{h_{2}^{2}}\right)&- \frac{1}{h_{1}^{2}}\\ &&&-\frac{1}{h_{1}^{2}}&2\left(\frac{1}{h_{1}^{2}}+\frac{1}{h_{2}^{2}}\right) \end{pmatrix} , \tag{22}\]  
\[\mathbf{D}= \begin{pmatrix}-\frac{1}{h_{2}^{2}}\\ -\frac{1}{h_{2}^{2}}\\ &\ddots\\ &&&-\frac{1}{h_{2}^{2}}\end{pmatrix}  \tag{23}\]  
\[\mathbf{const}_{j}= \begin{pmatrix}const\big{(}x_{1},y_{j}\big{)}\\ const\big{(}x_{2},y_{j}\big{)}\\ \vdots\\ const\big{(}x_{N-2},y_{j}\big{)}\\ const\big{(}x_{N-1},y_{j}\big{)}\end{pmatrix}  \tag{24}\]  
Note that the first and last terms of vector \(const_{j}\) already contain the boundary conditions at \(x=a\) and \(x=b\). Equation (21) can also be expanded to the following matrix form similar to Equation (25), as  
\[ \begin{pmatrix}\mathbf{C}&\mathbf{D}\\ \mathbf{D}&\mathbf{C}&\mathbf{D}\\ &&\ddots\\ &&\mathbf{D}&\mathbf{C}&\mathbf{D}\\ &&&\mathbf{D}&\mathbf{C}\end{pmatrix}\begin{pmatrix}\mathbf{\phi}_{1}\\ \mathbf{\phi}_{2}\\ \vdots\\ \mathbf{\phi}_{M-2}\\ \mathbf{\phi}_{M-1}\end{pmatrix} = \begin{pmatrix}\mathbf{const}_{1}-\mathbf{D}\mathbf{\phi}_{ 0}\\ \mathbf{const}_{2}\\ \vdots\\ \mathbf{const}_{M-2}\\ \mathbf{const}_{M-1}-\mathbf{D}\mathbf{\phi}_{M}\end{pmatrix}