Real-Time Stress Field Prediction of Umbilical Based on PyEf-U-Net Convolutional Neural Network

Abstract

Stress field analysis is an essential part of umbilical component layout design. The stress field analysis of an umbilical, via numerical simulation, has commonly been applied in practical engineering. The high economic and time cost associated with numerical simulation and analysis of the stress field in an umbilical has been replaced by data-driven, deep-learning-based, real-time computational methods. In this study, a novel Pyramidal Efficient U-Net (PyEf-U-Net) network is proposed to predict the stress field distribution of the umbilical. The input dataset is obtained via the Differential Evolution-Generalized Lagrange Multiplier (DE-GLM) method, which is entered into the network for training, with a detailed discussion of the effects of hyperparameters such as optimizer, learning rate, and loss function on the performance of the network. The experimental research demonstrates that the proposed PyEf-U-Net can accurately predict the stress field of the umbilical in real time with a prediction accuracy of 94.2%, which is superior to other deep learning networks. The proposed method can provide an effective way for rapid mechanical analysis and design of the umbilical in practical engineering, while the method can be extended to the mechanical analysis and design of other similar marine engineering equipment structures.

1. Introduction

An umbilical is the core equipment of the entire production system, which is widely applied to connect a floating body to the subsea manifold, providing hydraulic transmission, chemical injection, electrical energy, and control signal transmission for sub-sea trees. The subsea production system’s umbilical structure is shown in Figure 1 [1]. The umbilical is typically a multi-layer elongated helical wound armored structure, which generally consists of internal components such as cables, optical cables, steel pipes, etc. During the operation of the umbilical, it is subject to complex loads such as waves and currents. Hence, the safety of the umbilical structure has been directly related to the normal operation of the entire underwater production system. The mechanical properties of the umbilical can directly reflect the safety of the structure. Nevertheless, the umbilical consists of a multi-material and multi-component composite structure, with a large amount of nonlinear contact and friction, which is a typical heterogeneous anisotropic structure. The above factors bring great challenges in solving the mechanical properties of the umbilical [2,3].

During operation, the umbilical is subjected to tension at the top of the umbilical where it connects to the floating body, and the mechanical behavior of the umbilical at the touchdown point is also subjected to axial compression. Mechanical behaviors such as tension and compression act on the umbilical structure over the long term. Consequently, it is essential to analyze the mechanical properties of the umbilical. To effectively restore the working environment of the umbilical, the industry normally adopts Deepwater Immersion Performance (DIP) experiments that simulate the actual working conditions to perform relevant research. Braga et al. [4] reproduced the lateral slip instability damage pattern of the umbilical helical armor layer by the DIP experimental method, which revealed that the bending load plays an important role in inducing the lateral slip instability of the armor layer. Secher et al. [5] observed the lateral slip instability of the armor layer under combined bending and compression loads while conducting structural simulation certification experiments for the umbilical at 3000 m water depth, and then determined the limit capacity of the armor layer of the umbilical to resist lateral slip instability damage through experimental data analysis. Considering the extremely high cost and difficult operation of DIP experiments, most scholars have tried to investigate the umbilical from indoor equivalent loading experiments, theoretical analysis, and numerical simulations. Sousa et al. [6] constructed an indoor simulation experimental setup to simulate the umbilical structure with combined bending and compression loads and carried out the corresponding destabilization damage research. The experimental data on the displacement and torsion angle of the steel armor wire have been analyzed, which has verified the feasibility of the numerical simulation and explained the destabilization damage mechanism.

Sertã et al. [7] designed an experimental setup to simulate combined bending, compression, and external pressure loads, and performed instability damage simulation experiments for umbilical with 2.5-inch and 4-inch diameters, simultaneously comparing the traditional results with the data to prove the validity of the developed numerical simulation. Féret et al. [8] used a formula that allows a preliminary estimation of the service life of the umbilical by calculating the inter-layer slip and stresses caused by bending. Nevertheless, the method must be completed by experimentally determining the friction coefficient. Knapp [9] pioneered a new set of equilibrium equations that were used to solve the cable stiffness matrix under tension and torsion. Bahtui et al. [10] performed a numerical analysis of the umbilical, which compared the numerical simulation results with various load condition scenarios to demonstrate the feasibility of the numerical simulation. Guttner et al. [11] developed 2D and 3D models for umbilical subjected to crush load, which were used to predict the strain field of the umbilical with two different numerical methods. The results demonstrated that the numerical calculation of the 2D model can be faster under the crushing load, yet the calculation results of the two models have to be consistent. Qu et al. [12] designed a new structure for umbilicals to adapt to the working environment of 400 m water depth and utilized UFLEX to study the stress and mechanical properties of the umbilical under 400 m water depth, which verified the rationality and reliability of the structural design of the umbilical. Wagner et al. [13] developed a novel manufacturing process for seam-welded stainless steel pipes that improves the mechanical properties of the pipes, which avoids the effects of water depth, pressure, and temperature on the structural properties of the umbilical.

Yang et al. [14] proposed a semi-analytical calculation method to calculate the nonlinear tensile-torsional coupling effect of an umbilical. Tensile experiments were conducted on the umbilical, which demonstrated the accuracy of the method. Simultaneously, it can provide beneficial data references for the design of an umbilical. Mechanical properties of the umbilical are normally analyzed with theoretical analysis, experimental testing, and numerical simulation. Due to the existence of substantial nonlinearities in umbilical, it is difficult to accurately solve the mechanical behavior of the umbilical via theoretical analysis, while the experimental cost is extremely high. Consequently, numerical simulation [15] has become the mainstream method for the current analysis of the mechanical behavior of an umbilical. Nevertheless, numerical simulation computation is frequently time-consuming and prone to non-convergence.

Consequently, determining how to avoid the high cost of calculation time while ensuring the accuracy of the calculation results is essential. With the development of deep learning, considerable efforts have been focused on using deep learning for real-time computation [16,17,18,19,20,21], which is applied to avoid time cost consumption. Bhaduri et al. [22] applied deep learning methods to predict the stress fields of fiber-reinforced composites to address the huge computational time cost associated with finite element analysis. The stress field of the composites with different numbers of fibers has been obtained via the U-Net network, which provides a reference for the subsequent material stress analysis. Nie et al. [23] proposed two different deep learning models, SCSNet and StressNet, for predicting the stress field distribution in 2D cantilever beams, respectively.

Among the existing studies on the mechanical properties of umbilical, few studies have been conducted to obtain the stress field distribution of umbilical in real time. Vinicius et al. [24] combined finite element analysis with deep learning and proposed the NARX-CNN network model to predict the tension and curvature of the umbilical in the first time, and the accuracy of the proposed network model is higher than that of the finite element method, which avoids the disadvantage of the high computational cost of finite elements. Yan et al. [25] employed a residual neural network to predict the load–displacement curve of umbilical under radial compression, which prevented the computational errors due to the consideration of elastic deformation in the traditional theoretical calculation. The network can only predict the load–displacement curves of large-diameter umbilicals in the plastic stage with data from small-diameter umbilicals, and the accuracy of the network decreases with the reduction of input data. Although the above research has brought about the prediction of the mechanical properties of umbilical, the network structure is relatively simple. Aiming to avoid the time and economic cost consumption caused by traditional numerical simulation methods, a novel deep convolutional neural network PyEf-U-Net is proposed. Adding the pyramidal convolution (Pyra-Con) module in the network transition layer, which employs multiple convolutions of different sizes for feature extraction of the feature map, which fuses multi-scale features. Additionally, adding the efficient channel attention module (EffCA) module after the low-level feature map, avoids the inefficiency of dimensionality reduction operations for channel attention prediction, while capturing feature information across channel interactions in an extremely lightweight manner. The results demonstrate that the PyEf-U-Net not only has fast convergence characteristics and well-generalized capability but also can achieve real-time accurate prediction of the mechanical properties distribution of umbilical.

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 $P_i^s=\left(x_i,y_i\right)$ 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:

$$G_E={\lambda }_E\pi R^2$$

(1)

$$C_E={\eta }_E\cdot G_E$$

(2)

$$M_E=\frac{{\zeta }_E}{G_E}$$

(3)

where λ_E refers to the geometric coefficient, η_E refers to the cost coefficient, and ζ_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:

$$f_1=\min G_E$$

(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:

$$\mathit{G}=E_i{A}_i$$

(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 $\mathsf{\Delta }$. The size of the distance $\mathsf{\Delta }$ can represent the mechanical properties of the umbilical. $\mathsf{\Delta }$ and the mechanical properties of the umbilical M_B can be expressed as

$$\mathsf{\Delta }=\frac{{\mathit{G}}_i^S{P}_i^S+{\mathit{G}}_j^C{P}_j^C}{{\displaystyle \sum _{i=1}^m{\mathit{G}}_i^S+{\displaystyle \sum _{j=1}^n{\mathit{G}}_j^C}}}$$

(6)

$$M_B=\frac{{\xi }_B}{\mathsf{\Delta }}$$

(7)

where ${\mathit{G}}_i^S$ denotes the pseudo-gravity of the i-th steel pipe and ${\mathit{G}}_j^C$ denotes the pseudo-gravity of the j-th cable.

The center deviation distance varies when the layout form of the umbilical varies. When $\mathsf{\Delta }\ne 0$, the umbilical layout is unbalanced; when $\mathsf{\Delta }=0$, the umbilical layout is balanced. Consequently, the optimization objective 2 is

$$f_2=\min \mathsf{\Delta }$$

(8)

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

$$D_f={\displaystyle \sum _{\begin{array}{l}i=1,j=1\\ i\ne j\end{array}}^n{\psi }_{f}f\left({\left(2R^S\right)}^2-{‖P_i^S-p_j^S‖}^2\right)}$$

(9)

in which ${\psi }_f$ is the coefficient of mechanical properties. Hence, optimization objective 3 is expressed as

$$f_3=\min D_f$$

(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

$$\min f=K_1{f}_1^{’}+K_2{f}_2^{’}+K_3{f}_3^{’}$$

(11)

where K₁, K₂, and K₃ 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:

$$\left\{\begin{array}{l}{‖P_i^S-P_j^S‖}^2-{\left(2R^S\right)}^2\ge 0;i\ne j;i,j=1,2,\cdot \cdot \cdot ,m\\ {‖P_i^C-P_j^C‖}^2-{\left(2R^C\right)}^2\ge 0;i\ne j;i,j=1,2,\cdot \cdot \cdot ,n\\ {\left(R-R^S\right)}^2-{‖P_i^S‖}^2\ge 0;i=1,2,\cdot \cdot \cdot ,m\\ {\left(R-R^C\right)}^2-{‖P_j^C‖}^2\ge 0;j=1,2,\cdot \cdot \cdot ,n\\ {‖P_i^C-P_j^S‖}^2-{\left(R^C+R^S\right)}^2\ge 0;i=1,2,\cdot \cdot \cdot ,m;j=1,2,\cdot \cdot \cdot ,n\end{array}\right.$$

(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 K₁ = K₂ = K₃ = 1/3, the optimization model can be expressed as

$$\begin{array}{l}find:\mathit{X}=\left[\left(x_i,y_i\right),\left(x_j,y_j\right)\right]\left(i=1,2,\cdot \cdot \cdot ,m;j=1,2,\cdot \cdot \cdot ,n\right)\\ \min :f\left(\mathit{X}\right)=\frac{1}{3}{f}_1^{’}+\frac{1}{3}{f}_2^{’}+\frac{1}{3}{f}_3^{’}\\ s.t.:\left\{\begin{array}{l}{\left(2r\right)}^2-{\left(x_i-x_j\right)}^2-{\left(y_i-y_j\right)}^2\le 0;i\ne j;i,j=1,2,\cdot \cdot \cdot ,m+n\\ x_i^2+y_i^2-{\left(R-r\right)}^2\le 0;i=1,2,\cdot \cdot \cdot ,m+n\end{array}\right.\end{array}$$

(13)

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. The DE algorithm mainly consists of four steps:

(1)

Initialize the Population

The size of the population size in the DE algorithm is denoted by NP, while the size of the dimension of the individuals is denoted by D, and the initialized individual is expressed as

$$x_{ij}\left(0\right)=x_{ij}^L+r\times \left(x_{ij}^U-x_{ij}^L\right)$$

(14)

in which, i = 1, 2, ⋯, NP, j = 1, 2, ⋯, D. x_ij represents the number of dimensions of the population individuals, while the upper and lower limits are represented by U and L, respectively. r refers to the random number between [0,1].

(2)

Variant Operation

To maintain the diversity of the population, the DE/rand/1/bin difference strategy is used to make the population variable. The expression is

$$v_{ij}\left(n+1\right)=x_{r1}\left(n\right)+Q\cdot \left(x_{r2j}\left(n\right)-x_{r3j}\left(n\right)\right)$$

(15)

where the scaling factor Q is a constant, and r₁, r₂, and r₃ represent the individual number.

(3)

Crossover Operation

Crossover operation is introduced to maintain the diversity of the population. The crossover operation is

$$u_{ij}\left(n+1\right)=\left\{\begin{array}{l}{w}_{ij}\left(n+1\right),rand\left(j\right)\le CR\stackrel{}{}or\stackrel{}{}j=k\\ x_{ij}\left(n\right),otherwise\end{array}\right.$$

(16)

in which CR denotes the crossover probability and $rand\left(j\right)\in \left[0,1\right]$.

(4)

Selection Operation

The DE algorithm ensures the convergence of the computation according to the adaptation evaluation mechanism, which is operated as follows:

$$x_{ij}\left(n+1\right)=\left\{\begin{array}{l}{u}_i\left(n+1\right),f\left(u_i\left(n+1\right)\right)<f\left(x_i\left(n\right)\right)\\ x_i\left(n\right), f\left(u_i\left(n+1\right)\right)\ge f\left(x_i\left(n\right)\right)\end{array}\right.$$

(17)

where $u_i\left(n+1\right)$ represents the experimental individual and $x_i\left(n\right)$ denotes the baseline individual.

Penalty functions [27] can be used with the DE algorithm to solve constrained optimization problems. The general form of the penalty function is given as

$$P\left(X\right)=f\left(X\right)+{\displaystyle \sum _{i=1}^n{r}_i{G}_i\left(X\right)}$$

(18)

$$G_i\left(X\right)=\max \left[0,g_i\left(X\right)\right]$$

(19)

where f(X) denotes the objective function, $r_i$ stands for the penalty coefficient, G_i(X) represents the constraint violation function, and g_i(X) denotes the constraint function.

GLM [28] combines the Lagrange multiplier method and the outer point penalty function method for solving the optimal value, which is one of the ways to solve constrained optimization problems. After obtaining the feasible solution $X^1$ with the DE algorithm, the exact solution $X^2$ is then rapidly acquired via the GLM algorithm, which can be simplified as follows:

$$\left\{\begin{array}{l}\min f\left(X^1\right)\\ s.t.:g_i\left(X^1\right)\le 0\begin{array}{cc}& i=1,2,\cdot \cdot \cdot ,n\end{array}\end{array}\right.$$

(20)

in which f(X¹) represents the objective function and g_i(X¹) refers to the inequality-constrained equation.

Four different layout cases of the umbilical are presented in Figure 4.

3. Numerical Simulation and Experimental Verification

The typical umbilical cross-section is shown in Figure 5, which is mainly composed of six steel pipes, three cables, one optical cable, an inner sheath, armor wires, an outer sheath, and filling bodies. The geometric size of each component of the umbilical is shown in

Table 1. Umbilical structure size parameters [14].

Component	Material	Quantity	Structure Size
			Inner Diameter (mm)	Thickness (mm)	Outer Diameter (mm)
Steel pipe	Duplex stainless steel pipe	6	12.7	1.0	14.7
	Polyethylene sheath	6	14.7	1	16.7
Cable	Copper conductor	3 × (4 × 7 mm2)			3.15
	Polyethylene sheath	3	30.1	1.5	33.1
	XLPE insulation	12	3.15	0.725	4.6
Optical cable	Seamless steel	1	2.6	0.2	3.0
	Inner wire layer	7	3.0	2.0	7.0
	Outer wire layer	13	7.0	2.0	11.0
	Polyethylene sheath	1	11.0	2.5	16.0
Armor wire	Inner layer	46	74.0	5	84.0
	Outer layer	52	84.0	5	94.0
Inner sheath	Polyethylene	1	68.0	3.0	74.0
Outer sheath	Polyethylene	1	94.0	5.0	104.0

. Due to the multi-component and multi-material characteristics of the umbilical, a large amount of contact and friction exists between the components, which can cause great difficulties when performing numerical simulations. Consequently, it has been found in existing studies [29,30,31,32,33] that certain parts are typically simplified in modeling due to the smaller contribution of these components to the overall strength of the umbilical. Cable is commonly simplified to the polymer dense-wrapped copper core structure, while the optical cable is simplified to polymer dense-wrapped steel shell structure. The simplified umbilical is shown in Figure 6.

Numerical simulation analysis of the umbilical is performed via ABAQUS simulation software, which is compared and analyzed with subsequent experiments. The total length of the model is the same as the real size, with a total length of 200 mm. The mesh type, C3D8R, in ABAQUS, is chosen since the precision of the computation analysis is not greatly affected when deformation distortion occurs in the mesh. Since the internal components of the umbilical are mostly helical wound structures, the swept mesh division method is adopted to arrange the same mesh seeds in the axial direction, which makes the different components of the umbilical maintain the same size in the axial direction and ensures better mesh contact between adjacent layers. The whole mesh of the umbilical model is shown in Figure 7. The mesh independent validation is shown in

Table 2. The mesh independent verification.

Number of Mesh	Displacement	Error
149,360	0.2827 mm	5.7%
150,960	0.2675 mm	---
162,640	0.2608 mm	2.5%

. The number of mesh was selected as 150,960 for calculation after considering the calculation accuracy and calculation time of the model.

Umbilical components consist of complex materials, and during the numerical simulation, the steel pipe and armored steel wire are made of stainless steel and low-carbon steel, respectively [34]. The sheath and filling body are made of high-density polyethylene material, and the conductor in the cable is made of copper material [35], with the material parameters shown in

Table 3. Material properties of the umbilical [36].

Material	Density	Elastic Modulus	Poisson’s Ratio
HDPE	1000 kg/m3	1.5 GPa	0.48
Copper	8900 kg/m3	108 GPa	0.33
Stainless steel	7800 kg/m3	210 GPa	0.3
Low carbon steel	7800 kg/m3	210 GPa	0.3

.

Umbilical is typically a helical wound structure, where the structure causes substantial contact and friction between the umbilical components. Umbilical components are large in number and contact each other, not only between neighboring components in the same layer in the circumferential direction but also between neighboring layers in the radial direction. During numerical simulation, generic contact in ABAQUS simulation software is employed to automatically identify all contact pairs. The normal behavior of the contact pairs adopts the default hard contact, which does not allow the contact surfaces to penetrate each other. The tangential behavior of the contact pairs is modeled by defining the Coulomb friction coefficient to simulate interlayer friction.

The boundary condition for the ABAQUS numerical simulation is shown in Figure 8. Two reference points, RP-1 and RP-2, are established in ABAQUS, which are connected, respectively, to the two ends of the umbilical by the way of motion coupling. Apply load to RP-1 and constraint to RP-2. Since the non-linear characteristics of the umbilical are obvious during the stressing process, the ABAQUS/Explicit solver is adopted for the numerical simulation calculation, which calculates the entire process of force change of the umbilical via the dynamic display.

To demonstrate the accuracy of ABAQUS numerical simulation, the mechanical properties of the umbilical have been evaluated with a servo control universal material testing machine. The testing machine model is GP-TS2000 M/300 KN, and the fixture filling case for the experiment is shown in Figure 9. The total length of the experimental sample umbilical is 200 mm, which is the same as the total length of the ABAQUS numerical model. The displacement of the umbilical at pressures of 35 KN and 10 KN, respectively has been tested, which compared the obtained experimental results with the ABAQUS numerical simulation results, as shown in

Table 4. Comparison of experimental and numerical simulation results.

	Experiment	Numerical Simulation	Experiment	Numerical Simulation
Force	35 KN	35 KN	10 KN	10 KN
Displacement	278.6 μm	267.5 μm	10.5 μm	10.9 μm

. It can be seen that the errors of numerical simulation results and experimental results are 4.15% and 3.67%, respectively, with the obtained results being in excellent agreement with the experimental results, which demonstrates the accuracy of the numerical simulation results.

4. PyEf-U-Net for the Stress Field Prediction

4.1. PyEf-U-Net Model

A novel convolutional neural network model is proposed to predict the layout of the umbilical stress field in real-time. The entire architecture of the proposed network model is illustrated in Figure 10. The proposed PyEf-U-Net deep learning model is composed of a symmetric structure with two parts: the encoder part and the decoder part. The overall network consists of eight convolutional units, 4 pooling layers, 4 EffCA modules, 1 Pyra-Con module, and multiple up-sampling blocks. Each convolutional block of the encoder consists of Dropout, batch normalized (BN) layer, and ReLU. The down-sampling operation is performed at a step size of 2, utilizing the maximum pooling operation, which halves the feature map resolution size and doubles the number of channels. The up-sampling operation in the decoder restores the feature map resolution with 2 × 2 transposed convolution, which doubles the number of channels, and the skip connection connects the corresponding feature map of the encoder. The Pyra-Con module is adopted to the last transition layer in the encoder and decoder, which adopts multi-branch networks with different convolutional kernels for multi-scale feature fusion to ensure computational efficiency. Additionally, the addition of the EffCA module follows the network’s low-level feature maps, which guarantee the network performance while reducing the network complexity with appropriate cross-channel interactions. Additionally, the addition of the EffCA module after the network’s low-level feature maps guarantees the network’s performance with appropriate cross-channel interactions. The input of the model is the different layout forms of the umbilical components. After a series of convolution, pooling, feature extraction, and merging operations, the output of the model is the corresponding stress field distribution.

4.2. EffCA Module

The efficient channel attention module (EffCA), after being applied to the low-level feature map of the PyEf-U-Net model, which brings about dramatic improvements of network performance with the introduction of fewer parameters. As shown in Figure 11, (a) represents the classical SE model and (b) denotes the EffCA model. In particular, the dimensionality reduction operation employed in the SE model has no significant positive effect on the prediction of channel attention. Nevertheless, the EffCA model, with its lightweight and efficient network structure, first compresses the spatial information of the features by using the global average pooling operation. Then, the convolution operation with a convolution kernel size of 1×1 is utilized to learn the channel attention information of the features, which is spliced and fused with the original feature information to output the final feature result. Compared with the SE model, the proposed structure can reduce the impact of the dimensionality reduction operation on the overall performance of the network, which can accurately and efficiently capture the cross-channel interactions of feature information.

4.3. Pyra-Con Module

A pyramidal convolution module (Pyra-Con) is adopted to the last transition layer of the PyEf-U-Net by adopting group convolution in each layer, and each branch using filters of different sizes and depths. It is shown in Figure 12, (a) denotes the standard convolution and (b) refers to the proposed pyramidal convolution. The pyramidal convolution module contains multiple convolutional kernels in parallel, from top to bottom, and the size of the convolutional kernels becomes progressively smaller and the depth of the convolutional kernels becomes progressively larger. The smaller convolutional kernel with a relatively small receptive field facilitates obtaining the small target and local detail information. While the larger convolutional kernel with the larger receptive field facilitates obtaining the large target and global semantic information. The input feature map is divided into different groups, with different groups applying separate kernels.

5. Experimental Results and Discussion

5.1. Experimental Parameters and Details

The dataset consists of 386 umbilical component layout results obtained with the DE-GLM algorithm. Since the prediction accuracy of the deep learning model depends largely on the training dataset samples, the selected data samples are representative of the umbilical component layout results, so that the PyEf-U-Net model can have a better generalization ability, which facilitates obtaining a high-performance deep learning model and acquiring the best model training parameters so that it can gain high-accuracy prediction results. A total of 193 data samples are selected from the original dataset as the test dataset, and the remaining original data samples are expanded via data augmentation methods. The expanded training dataset consists of 1544 data samples, and the validation dataset consists of 193 data samples. To clearly illustrate the idea of dataset division, part of the training dataset, validation dataset, and testing dataset are divided as shown in Figure 13.

The PyEf-U-Net network model is trained with Pytorch framework, Adam is chosen as the optimizer, the learning rate is set to 0.0001, and the Batch size is set to 4. The above parameters are the optimal hyperparameters for network training.

The loss function refers to the function of the cost required to map one event to another event associated with it, connecting the learning problem to the training network by minimizing the loss function, which achieves the purpose of training the network parameters. The trained network model has the best performance and the final prediction map of the stress field of the umbilical is closest to the sample labels. Multi-class cross entropy loss function is chosen in this paper to train the proposed PyEf-U-Net network, and the equation is shown below:

$$Loss\left(\mathrm{CrossEntropyLoss}\right)=-\frac{1}{N}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^K{y}_{i,j}\log {\widehat{y}}_{i,j}}}$$

(21)

where $y_{i,j}$ denotes the data label value, ${\widehat{y}}_{i,j}$ refers to the model output value, N refers to the number of samples, and K indicates the label value.

To evaluate the reasonability of the prediction results of the network model, the calculation results of the DE-GLM algorithm are compared and analyzed with the prediction results of the network model, with the formula shown below:

$$Accuracy=\frac{TP+TN}{TP+FP+FN+TN}$$

(22)

$$F_1=2\times \frac{Precision\times Recall}{Precision+Recall}$$

(23)

$$Precision=\frac{TP}{TP+FP}$$

(24)

$$Recall=\frac{TP}{TP+FN}$$

(25)

The classification confusion matrix is shown in

Table 5. Classification confusion matrix.

Sample	Positive	Negative
Positive	TP	FN
Negative	FP	TN

.

5.2. Analysis of Network Model Parameters and Discussion of Prediction Results

The performance of the proposed PyEf-U-Net network model is validated to be advantageous by comparing it with the other three deep learning network models, as shown in Figure 14. It can be seen that the USE-Net [37] and the SCSE-U-Net [38] have similar change patterns of loss value curves; however, both appear significant abrupt changes during the iterations, demonstrating the poor stability of the models, which causes low accuracy of the final prediction results and cannot achieve the final prediction purpose. Moreover, the value of loss declines slowly in the first 1500 iterations and the convergence speed of both deep learning models becomes slower. Additionally, although the U-Net [39] has no mutations throughout the training process, the network converges slowly and cannot adapt well to the dataset. By comparison, the loss–value curve of the PyEf-U-Net changes smoothly, the minimum loss value can be obtained without multiple iterations and gradually stabilizes, while the overall convergence speed of the network is faster than the other three network models. Consequently, the proposed PyEf-U-Net has superior performance than the other three deep learning network models.

The stress field distribution of the umbilical with different layout forms is applied to verify the performance of the PyEf-U-Net, and the stress field prediction results as shown in

Table 6. Comparative analysis of stress field prediction results for umbilical.

	F = 4 MPa	F = 4 MPa
Numerical simulation
U-Net
USE-Net
SCSE-U-Net
PyEf-U-Net
Numerical simulation
U-Net
USE-Net
SCSE-U-Net
PyEf-U-Net

. Compared with the numerical simulation results, the prediction results obtained from the PyEf-U-Net proposed have the best agreement with the numerical simulation results, while the other three network models present the problem of the imprecise numerical prediction of the stress field of the umbilical. It can be observed that U-Net obtained the worst prediction results, while both USE-Net and SCSE-U-Net outperformed U-Net, which is due to the fact that the mentioned deep learning model adding the attention module to the U-Net network, capturing feature information from the channel dimension, assigning higher weights to important feature regions, and suppressing irrelevant features. Nevertheless, the prediction results of the above network are frequently unsatisfactory in places where the stresses in the umbilical are concentrated. On the contrary, the proposed PyEf-U-Net can successfully avoid the above-mentioned situations and well distinguish the stress cases in different regions. More importantly, adding the EffCA module and Pyra-Con module to the PyEf-U-Net has combined the ability of the lightweight EffCA module to capture channel dependencies with the ability of the Pyra-Con module to fuse multi-scale features, which enables the prediction results of the PyEf-U-Net to be closest to the numerical simulation results with the highest accuracy, particularly in regions such as the umbilical stress concentration, where the numerical prediction of the boundary stress can be accurate and detailed, which can yield the best prediction results of the stress field.

The learning rate parameter determines whether the loss function can converge, which in turn affects the accuracy of the prediction results of the deep learning network. To investigate the effect of the learning rate on the PyEf-U-Net deep learning model, the learning rate is set to 0.01, 0.001, 0.0001, and 0.00001. Comparing the change of loss values of the PyEf-U-Net deep learning model with different learning rates, as shown in Figure 15. At learning rates of 0.01 and 0.001, the loss value decreases to around 0.2 in the early stage of iterative training; however, the network fluctuates drastically during the later iterations and the stability of the network is poor. It indicates that when the learning rate is 0.01 and 0.001, the network cannot consistently and steadily learn the feature information from the original dataset and has poor learning ability. Additionally, with the learning rate of 0.00001, the network performance is the worst, the loss value iterations converge slowly and eventually stay fluctuating around 0.3, which has no optimal learning ability. In comparison, with the learning rate of 0.0001, the network rapidly decreases and gradually stabilizes the loss value at the beginning of training, which converges to the minimum value of around 0.04, with no sudden mutations occurring during the entire process. Consequently, 0.0001 is selected to be the learning rate parameter for the PyEf-U-Net network.

Batch size has an impact on the accuracy of the stress field prediction results of the umbilical. Comparative analysis for different batch sizes is performed, and the umbilical prediction results are shown in Figure 16. With batch sizes 2, 8, and 16, inaccurate pre-dictions of the stress field at the boundary of the umbilical can occur, especially in the region of stress concentration. Through comparison and analysis, it was found that when the batch size is 4, the above existing problems can be successfully avoided, which can ac-curately predict the area where the stress of the umbilical concentration is. Consequently, the batch size parameter was eventually chosen to be 4.

The optimization algorithm is the significant network parameter that affects the iterative training of deep learning models, whose selection directly influences the performance of the PyEf-U-Net deep learning. To investigate the effects of different optimization algorithm selections for the PyEf-U-Net deep learning model, four different types of optimizers are chosen for comparative analysis, as shown in Figure 17. It can be seen that the SGD has the worst performance with slow iteration of the loss value curve and the loss value decreases slowly, which results in the poor learning ability of the network. This is due to the fact that SGD is prone to training in the direction of deviating from the optimal network strength and the network training is unstable. Nevertheless, with regard to the Adagrad optimizer, due to the inclusion of the momentum matrix, the network is able to adapt to changes in the learning rate, which is overall superior to the curve of variation in the loss values of SGD. Nevertheless, the accuracy of the PyEf-U-Net cannot be guaranteed due to the high loss values during the late iterations. Additionally, the RMSProp updates the parameters through adjusting the learning rate, and its loss value change is superior to that of the SGD and Adagrad; however, the algorithm has a slower loss value decrease and slower convergence in the pre-iterative training period, with significant abrupt changes during the late stage of iteration. By comparison and analysis, the loss value variation curves obtained by the Adam significantly outperformed those obtained by the other three optimization algorithms. Rapid decrease and gradual stabilization of the loss value in the pre iterative training period, stabilizing at about 0.04, which is substantially lower than the loss values obtained by the other three optimization algorithms, simultaneously, without any sudden fluctuation during the whole network training process. Hence, the Adam has the best performance.

The loss function is commonly associated with optimization problems as a learning criterion, i.e., the performance of the model prediction is measured by minimizing the loss function. The variation of loss values with different loss functions during the iterative training of the network is shown in Figure 18. The NLLLoss2d loss function is prone to sudden changes during the 1500–2400 network iterations, and the low learning efficiency of the PyEf-U-Net deep learning network leads to unsatisfactory prediction results. Nevertheless, although the Multiclass Dice loss functions smoothly during the early iterations, obvious fluctuations exist during the later period, with unstable model training and slowly decreasing loss values during the early period. Additionally, the Cross Entropy loss function selected has the most satisfactory performance, which can not only successfully avoid the shortcomings of the NLLLoss2d loss function and Multiclass Dice loss function but also has the characteristics of fast convergence, no sudden change, and stability, which gives the proposed network model a strong learning ability, with the best network performance for umbilical stress field prediction, which provides the accurate prediction of output results. Consequently, in comparison, the Cross Entropy loss function was chosen from the PyEf-U-Net deep learning network.

As shown in

Table 7. Analysis of ablation experiment results.

Basic Model	EffCA	Pyra-Con	Accuracy
$\surd$			0.854
$\surd$	$\surd$		0.882
$\surd$		$\surd$	0.895
$\surd$	$\surd$	$\surd$	0.942

, the effect of the combination of four different modules on the accuracy of the network model is investigated. The Basic model in the first row refers to the U-Net without any modules, which obtains the accuracy of 0.854. The second row of the table refers to the accuracy of 0.882, only adding the EffCA module to the Basic model. The third row of the table refers to adding only the Pyra-Con module to the Basic model, which yields an accuracy of 0.895. Nonetheless, the last row of the table refers to the PyEf-U-Net, which adds the EffCA module and Pyra-Con module to the low-level feature map and the last transition layer of the network, respectively, in comparison with the highest accuracy rate of 0.942. It has been experimentally demonstrated that each module is essential to improving the prediction accuracy, and the PyEf-U-Net can only obtain the highest accuracy by adding both the EffCA module and Pyra-Con module, which can achieve high-accuracy umbilical stress field prediction results.

The comparison of the time used for the numerical simulations with different layouts of the umbilical and the predicted time of the PyEf-U-Net is shown in

Table 8. Calculation time analysis of stress field for network model and numerical simulation.

Cross-Sectional Layout	Simulation Software Time Consumption	PyEf-U-Net Time Consumption
	30 min and 15 s	3.877 s
	51 min and 12 s	3.582 s
	34 min and 40 s	4.037 s
	28 min and 10 s	2.866 s
	10 min and 41 s	3.432 s

. It is evident that the numerical simulation time exponentially increases with increasing the number, type, and number of layers of the components. The variation of such parameters can lead to the explosion increasing numerical simulation time, which will be costly in terms of time. Nevertheless, the prediction time of the proposed PyEf-U-Net network in this paper ranges from 2–5 s, which improves the computational efficiency; the time cost is almost negligible, and the prediction time is not affected by the number, type, and layers of the components. Consequently, the method of using deep learning networks for stress field prediction of the umbilical will significantly improve the efficiency of the mechanical property analysis of the umbilical in the marine engineering field.

6. Conclusions

As an important equipment of marine engineering, it is essential to analyze the mechanical properties of the umbilical. A novel PyEf-U-Net convolutional neural network model has been proposed in this paper to overcome the drawbacks of the traditional finite element method, which is seriously time-consuming and hard to converge, and which achieves the purpose of real-time accurate prediction of mechanical properties of the umbilical. The experimental validation demonstrates that the proposed deep learning model can not only accurately predict the stress field layout of umbilical within 2–5 s, but also has an accuracy rate as high as 94.2% compared to other state-of-the-art deep learning models. Additionally, the effects of hyperparameters such as the optimizer, learning rate, batch size, and loss function of the PyEf-U-Net network model have been discussed in detail in the paper. Among the currently available studies, the feasibility of the proposed deep learning method has been verified with 2D arithmetic examples, which can accurately predict the stress field distribution of the umbilical in real-time, and which provides an effective way to predict other mechanical properties of the umbilical in real-time. Alternatively, the method can be extended to 3D models, which are not fundamentally different in concept and principle, except for the relatively high cost of the dataset collection process. In particular, the complex distribution of umbilical load forms in the marine environment will be an enormous challenge to predicting the mechanical properties of umbilical in real-time. In future work, more complicated 3D engineering cases will be utilized to input the network model, which will enhance the generalization ability of deep learning methods to predict the distribution of umbilical mechanical properties in real-time in a real marine environment. Simultaneously, it can be applied not only to the study of the mechanical properties of an umbilical but also extended to the analysis of the mechanical properties of similar structures such as cables.