# Real-Time Stress Field Prediction of Umbilical Based on PyEf-U-Net Convolutional Neural Network
## 2 Mathematical Modeling Solutions  
The mechanical properties of umbilical are greatly affected by the way in which their components are laid out. The existing umbilical design process largely depends on artificial experience, which enables the layout of the components to meet symmetry and compactness to the extent possible. To address the problems of blindness and inefficiency in artificial experience, as well as for the diversity of subsequent datasets, the DE-GLM algorithm is proposed to provide feasibility for the rapid design of umbilical component layouts. With certain geometric and mechanical conditions satisfied, the umbilical component layout problem can be regarded as an optimization problem. During the process of setting up the optimization model, it is necessary to make certain assumptions considering the complex diversity of the umbilical cross-section. (1) The cable and optical cable in the internal components have similar characteristics in terms of structural mechanical properties. Hence, during the optimization process, both are considered similar components, i.e., the optical cable is simplified to the cable. (2) During the optimization process, it is assumed that the components are rigid bodies and the mutual deformation between the components has not been considered. (3) Assuming the same size of different kinds of components during the optimization process, and ignoring the impact of filling bodies on the layout. After the layout form has been determined, the filling body can be added by itself according to the shape between the components. The coordinate details of the umbilical components are shown in Figure 2. The coordinates of the centers of the steel pipe and the cable are \(P_{i}^{s}=(x_{i},y_{i})\) and \(P_{j}^{C}=\left(x_{j},y_{j}\right)\), respectively. Throughout the optimization computation, it is necessary to let these components move in a circle of radius \(R\) until the optimal solution is obtained while the radius \(R\) is minimized. Compactness between the components improves the tensile strength of the umbilical. The umbilical is required to possess high tensile strength in the operating environment and it is necessary to make the components as compact as possible while reducing costs. It is assumed that the geometric index \(G_{E}\) of the compactness of the component layout of the umbilical is affected by the cross-sectional area. The umbilical geometric index \(G_{E}\) can directly affect the cost \(C_{E}\) and mechanical properties \(M_{E}\), which are related as shown below:  
\[G_{E}=\lambda_{E}\pi R^{2} \tag{1}\]  
\[C_{E}=\eta_{E}\cdot G_{E} \tag{2}\]  
\[M_{E}=\frac{\zeta_{E}}{G_{E}} \tag{3}\]  
Figure 2: Coordinates of the center position of each component of the umbilical [1]. where \(\lambda_{E}\) refers to the geometric coefficient, \(\eta_{E}\) refers to the cost coefficient, and \(\zeta_{E}\) refers to the mechanical property coefficient. From Equations (1)-(3), it can be seen that when \(G_{E}\) is smaller, the the cost of the umbilical is lower, and the mechanical properties simultaneously better. Consequently, optimization objective 1 is expressed as:  
\[f_{1}=\min G_{E} \tag{4}\]  
The umbilical mainly sustains the combination form of tensile loads and bending loads during operation. Tensile performance is more critical to the resistance of the umbilical to mechanical failure. Consequently, for better resistance to the failure of the mechanics, the pseudo-gravity index \(G\) is introduced, with the following expression:  
\[\mathbf{G}=E_{i}A_{i} \tag{5}\]  
where \(E_{i}\) represents the elastic modulus of the different components and \(A_{i}\) represents the area of the different components. As shown in Figure 3, the pseudo-gravity of the different components forms a set of parallel force systems, between the center of the parallel force system and the center of the umbilical cross-section, with the distance of size \(\Delta\). The size of the distance \(\Delta\) can represent the mechanical properties of the umbilical.