# Real-Time Stress Field Prediction of Umbilical Based on PyEf-U-Net Convolutional Neural Network
## 2 Mathematical Modeling Solutions
\(\Delta\) and the mechanical properties of the umbilical \(M_{B}\) can be expressed as  
\[\Delta=\frac{\mathbf{G}_{i}^{S}P_{i}^{S}+\mathbf{G}_{j}^{C}P_{j}^{C}}{\sum\limits_{i=1 }^{m}\mathbf{G}_{i}^{S}+\sum\limits_{j=1}^{n}\mathbf{G}_{j}^{C}} \tag{6}\]  
\[M_{B}=\frac{\zeta_{B}}{\Delta} \tag{7}\]  
where \(\mathbf{G}_{i}^{S}\) denotes the pseudo-gravity of the \(i\)-th steel pipe and \(\mathbf{G}_{j}^{C}\) denotes the pseudo-gravity of the \(j\)-th cable. The center deviation distance varies when the layout form of the umbilical varies. When \(\Delta\
eq 0\), the umbilical layout is unbalanced; when \(\Delta=0\), the umbilical layout is balanced. Consequently, the optimization objective 2 is  
\[f_{2}=\min\Delta \tag{8}\]  
\[G_{E}=\lambda_{E}\pi R^{2} \tag{1}\]0 Diagram of the pseudo-gravity of the components of the umbilical [1]. Contact wear between steel pipes should be avoided as much as possible during the layout of the umbilical components. The mechanical damage \(D_{f}\) between the steel pipes is expressed as  
\[D_{f}=\sum_{ \begin{array}{c}i=1,j=1\\ i\
eq j\end{array} }^{n}\psi_{f}f\left(\left(2R^{S}\right)^{2}-\left\|P_{i}^{S}- p_{j}^{S}\right\|^{2}\right) \tag{9}\]  
in which \(\psi_{f}\) is the coefficient of mechanical properties. Hence, optimization objective 3 is expressed as  
\[f_{3}=\min D_{f} \tag{10}\]  
The layout of the components is the multi-objective optimization problem. The optimal solution of multi-objective optimization problems cannot be optimal for all objectives simultaneously, which requires considering all problems in combination to obtain a compromise solution. To satisfy the requirements of layout with compact, balanced, and reduced wear, the multi-objective problem has been transformed into the single-objective problem via weighting coefficients, and the final optimization objective is expressed as  
\[\min f=K_{1}f_{1}^{{}^{\prime}}+K_{2}f_{2}^{{}^{\prime}}+K_{3}f_{3}^{{}^{\prime}} \tag{11}\]  
where \(K_{1}\), \(K_{2}\), and \(K_{3}\) denote the normalized weighting coefficients, and \(K_{1}+K_{2}+K_{3}=1\). \(f\) denote to the objective function. Accordingly, the optimization constraint condition is shown below:  
\[\left\{ \begin{array}{l}\left\|P_{i}^{S}-P_{j}^{S}\right\|^{2}-\left(2R^{S} \right)^{2}\geq 0;i\
eq j;i,j=1,2,\cdot\cdot\cdot,m\\ \left\|P_{i}^{C}-P_{j}^{C}\right\|^{2}-\left(2R^{C}\right)^{2}\geq 0;i\
eq j;i,j=1,2, \cdot\cdot\cdot,n\\ \left(R-R^{S}\right)^{2}-\left\|P_{i}^{S}\right\|^{2}\geq 0;i=1,2,\cdot\cdot \cdot,m\\ \left(R-R^{C}\right)^{2}-\left\|P_{i}^{C}\right\|^{2}\geq 0;j=1,2,\cdot\cdot \cdot,n\\ \left\|P_{i}^{C}-P_{j}^{S}\right\|^{2}-\left(R^{C}+R^{S}\right)^{2}\geq 0;i=1,2, \cdot\cdot\cdot,m;j=1,2,\cdot\cdot\cdot,n\end{array} \right. \tag{12}\]  
According to the assumptions that the cable and steel pipe sizes are the same (\(R^{S}=R^{C}=r\)), while the multi-objective weight coefficients are treated as \(K_{1}=K_{2}=K_{3}=1/3\), the optimization model can be expressed as  
\[find:\mathbf{X}=\left[\left(x_{i},y_{i}\right)\left(x_{j},y_{j} \right)\right]\left(i=1,2,\cdot\cdot\cdot,m;j=1,2,\cdot\cdot\cdot,n\right) \tag{13}\] \[\min:f(\mathbf{X})=\frac{1}{3}f_{1}^{{}^{\prime}}+\frac{1}{3}f_{2}^{{} ^{\prime}}+\frac{1}{3}f_{3}^{{}^{\prime}}\] \[s.t. :\left\{ \begin{array}{l}\left(2r\right)^{2}-\left(x_{i}-x_ {j}\right)^{2}-\left(y_{i}-y_{j}\right)^{2}\leq 0;i\
eq j;i,j=1,2,\cdot \cdot\cdot,m+n\\ x_{i}^{2}+y_{i}^{2}-\left(R-r\right)^{2}\leq 0;i=1,2,\cdot\cdot\cdot,m+n\end{array} \right.\]  
To solve the umbilical layout optimization model, an easy-to-use, and rapidly reliable global optimization algorithm is required. The DE [26] has the characteristics of greater robustness and high optimization efficiency. Hence, the DE algorithm is introduced to calculate the multi-objective optimization of the umbilical layout.