Research on Predicting the Mechanical Characteristics of Deep-Sea Mining Transportation Pipelines

Abstract

Deep-sea mining, as a critical direction for the future development of mineral resources, places significant importance on the mechanical characteristics of its transportation pipelines for the safety and efficiency of the entire mining system. This paper establishes a simulation model of the deep-sea mining system based on oceanic environmental loads and the mechanical theory of deep-sea mining transportation pipelines. Through a static analysis, the effective tension along the pipeline length, the maximum values of bending moment, and the minimum values of bending radius are determined as critical points for the dynamic analysis of pipeline mechanical characteristic monitoring. A dynamic simulation analysis of the pipeline’s mechanical characteristics was conducted, and simulation sensor data were obtained as inputs for the prediction model construction. A prediction model of pipeline mechanical characteristics based on the BP neural network was constructed, with the model’s prediction correlation coefficients all exceeding 0.95, enabling an accurate prediction of pipeline state parameters.

1. Introduction

With the continual growth in global demand for mineral resources, deep-sea mining has become an important direction for resource development. The seabed is rich in mineral resources, including polymetallic nodules, cobalt-rich crusts, and hydrothermal sulfides [1,2]. However, its development faces numerous technical challenges. Among these, the deep-sea mining transportation system plays a crucial role in deep-sea mining. This system, responsible for transporting seabed ore or slurry to the surface, is a central hub directly affecting mining efficiency and system safety [3]. The deep-sea mining transportation system connects the mining vehicle, intermediate storage, and the surface mothership through pipelines, with seabed collected slurry transported via flexible soft hoses and lifting hard pipes [4]. In the deep-sea environment, the transportation pipelines are subjected to various loads, including their own weight, buoyancy, forces from waves and currents, resistance and pressure from seawater. Studying the force situation of transportation pipelines and effectively predicting other mechanical parameters based on existing monitoring data to ensure the pipelines operate within a safe load range has significant implications for designing mining operation schemes, mining vehicle path ranges, and pipeline material performance.

Due to the extreme complexity of the deep-sea environment, deep-sea mining experiments require significant financial and human resources and need to address many uncontrollable risks. Establishing mathematical models of ocean environments and system equipment through numerical simulation to study the stress conditions of transportation pipelines can yield vast amounts of simulation data for constructing predictive models of pipeline mechanical characteristics. Currently, numerical simulation methods for calculating pipeline stress primarily include the concentrated mass method [5], finite difference method [6], and finite element method [7].

In the study of lifting hard pipe mechanics, American scholars such as Felippa, Chung, and Whitney have employed the finite element method to analyze the kinematic and dynamic characteristics of a 5486 m long lifting hard pipe [8,9,10,11,12]. They used three-dimensional beam elements to build the model, which is capable of axial, bending, and torsional coupled deformation. The pipeline model includes 18 elements, 19 nodes, and 114 degrees of freedom. In subsequent studies by Chung et al., the finite element method was further applied to research the static and dynamic behavior of deep-sea mining lift pipes and their attachments [13,14,15]. To reduce the vibration effects of marine hydrodynamics on lifting pipes, Chung, Cheng, and others designed several elastic damping connection devices on the pipeline and modeled the pipeline system using the finite element method and mass–spring element method for static and dynamic analysis [16,17,18,19,20]. Moreover, Mustoe, Huttelmaier, Cheng, and others used the Discrete Element Method (DEM) for the kinetic analysis of deep-sea mining lift pipes under 2D axial and bending coupling. This discrete element pipe description method is based on a system composed of rigid bodies connected by concentrated axial and bending stiffness. Mustoe and others conducted four nonlinear dynamic analyses with a 2D DEM model of an 5486 m deep-sea mining pipeline [21,22]. Huttelmaier applied the Discrete Element Method to calculate a 1905 m long marine vertical pipeline, including axial and bending deformations, and compared the results with the finite element method and analytical method [23]. Sup Hong based on the concentrated mass method and incremental iterative method, conducted a three-dimensional dynamic analysis of the deep-sea subsea mining lift pipe system, completing axial forced vibration in a vertical state and three-dimensional nonlinear dynamic analysis within the time domain [24,25]. Wu employed the stochastic phase spectrum method to generate random wave forces and vortex-induced excitations acting on risers in the time domain. He proposed a parametrically excited top-tensioned riser model that was simultaneously subjected to random wave and vortex-induced excitations [26].

In the study of flexible hose mechanics, Ghadimi [27] proposed a method for calculating the spatial shape and stress state of flexible hoses. This mathematical model principle involves dividing the hose into several mass–spring segments to analyze the force and motion relationship of the segments. Ablow and Schechter [28] conducted a finite difference analysis on submarine cables, combined with the system dynamics characteristics to establish and solve the differential equations of the flexible cable. Bernitsas et al. derived the mathematical three-dimensional equilibrium equations for flexible hoses and utilized the U.L. method to solve the equilibrium equations through the three-dimensional large displacement finite element incremental method [29]. Nordgren utilized the finite time integration method to analyze the situation where one end of the flexible hose was fixed and the other end was freely falling, under the assumption that the axial and shear effects of the hose were neglected [30]. Based on the static characteristics of the hydraulic conveying pipeline, Xu obtained a three-dimensional spatial model of the hose and conducted numerical simulations on the static displacement of the hydraulic lifting pipeline under steady-state conditions [31]. Wang utilized the OrcaFlex software to simulate the laying of a flexible pipeline with an outer diameter of 352.42 mm on the seabed at a depth of 3000 m. The study investigated the axial tension, bending moment, and stress–strain dynamics of the pipeline during the laying process and analyzed factors influencing the dynamic behavior of the pipeline [32].

Based on fundamental mechanics research, significant progress and breakthroughs have been achieved in recent years in the prediction and estimation of mechanical properties. Buljak, Maier, and others used inverse analysis to diagnose the mechanical properties and structures of materials, which aids in assessing potentially deteriorated material performance and mechanical damage within structures [33]. The combination of “drilling” methods with indentation tests and inverse analysis has been utilized to evaluate residual stress and mechanical properties, effectively reducing the additional damage and costs typically associated with conventional diagnostic procedures for structural components [34]. Zhang developed a stress prediction model based on synthesized nonlinear coefficients, which can effectively assess the stress state of specimens [35]. Song proposed a multimodal fusion prediction model that integrates material microstructure information and compositional data, providing a promising solution for the measurement and prediction of material properties [36].

Despite the rich force analysis provided by past research, there remains a significant challenge in force prediction, especially in quickly predicting under conditions of insufficient monitoring data. This paper aims to address this issue by establishing a simulation model of the deep-sea mining system and conducting static and dynamic simulation analyses on lifting hard pipes and hoses, further building a predictive model of pipeline mechanical characteristics based on the BP neural network, thereby providing guidance for the safety and reliability of the deep-sea mining transportation system.

2. Deep-Sea Mining Transportation Pipeline Mechanics Analysis

The deep-sea mining transportation pipeline, serving as a vital hub transporting seabed-collected polymetallic nodules to the surface and connecting the mining vehicle, intermediate storage, and mining ship, plays a critical role in system operation safety. To further investigate the motion laws and stress characteristics of the transportation pipeline in deep-sea mining operations and establish its mechanical prediction model, this section conducts static and dynamic analyses of the deep-sea mining transportation pipeline. This will provide a theoretical reference for subsequent simulation analysis and predictive model construction.

2.1. Static Analysis of Deep-Sea Mining Transportation Pipeline

The deep-sea transportation pipeline is considered a slender structure in marine engineering. Its static analysis usually employs the catenary equation [37]. The application of catenary theory [38] allows for a precise analytical calculation of the pipeline. Considering a differential segment, $ds$, the catenary model is illustrated in Figure 1. Here, $q\left(s\right)$ represents the gravitational force per unit length of the pipeline, $\theta$ denotes the angle between the pipeline unit’s tangent and the horizontal direction, and $T\mathit{sin}\theta$ and $T\mathit{cos}\theta$ are, respectively, the vertical and horizontal components of tension.

Based on the force balance analysis, the following is concluded:

(1)${\displaystyle \frac{d}{dx}}(T\mathit{cos}\theta )=0$

(2)${\displaystyle \frac{d}{dx}}(T\mathit{sin}\theta )dx-qds=0$

Assuming the constant $k=T\mathit{cos}\theta$, substituting $T=k/\mathit{cos}\theta$ into Equation (2) yields

(3)$k{\displaystyle \frac{d}{dx}}(\mathit{tan}\theta )=qds$

In the equation, $\mathit{tan}\theta ={\displaystyle \frac{dy}{dx}}$, and substituting this into Equation (3) yields

(4)$k{\displaystyle \frac{d^{2}y}{d{x}^2}}=qds$

Integrating Equation (4) yields

(5)$y={\displaystyle \frac{k}{q}}\mathit{cos}h\left({\displaystyle \frac{q}{k}}x+C_1\right)+C_2$

2.2. Dynamic Analysis of Deep-Sea Mining Transport Pipelines

This study is based on the Euler–Bernoulli beam theory [39] to analyze the dynamics of deep-sea transport pipelines, considering the coupling of the fluid and pipeline. It is assumed that the deformation of the transport pipeline is a small displacement, and the internal fluid is an ideal incompressible fluid. By analyzing the forces on the pipeline element $dz$, the motion control equations of the transport pipeline in a three-dimensional coordinate system are obtained. As shown in Figure 2, Figure 2a illustrates the force analysis on the pipeline wall, where $T$ is the axial tension of the pipeline, $F$ is the pressure of the internal fluid on the wall, $q$ is the frictional resistance of the internal flow on the wall, and $Q$ is the shear force of the pipeline. Figure 2b depicts the force analysis on the internal fluid of the pipeline.

The force balance equation for the small section of the pipeline wall in Figure 2a is as follows, where Equation (6) represents the Z-direction balance equation, and Equation (7) represents the Y-direction balance equation:

(6)${\displaystyle \frac{\partial T}{\partial z}}+qS-F{\displaystyle \frac{\partial y}{\partial z}}-m_{r}g-{\displaystyle \frac{\partial }{\partial z}}\left(Q{\displaystyle \frac{\partial y}{\partial z}}\right)+{\displaystyle \frac{\partial \left(A_e{P}_e\right)}{\partial z}}+{\rho }_{e}g{A}_{e}dz=0$

(7)$-C{\displaystyle \frac{\partial y}{\partial t}}-m_r{\displaystyle \frac{{\partial }^{2}y}{\partial t^2}}+F+qS{\displaystyle \frac{\partial y}{\partial z}}+{\displaystyle \frac{\partial Q}{\partial z}}+{\displaystyle \frac{\partial }{\partial z}}\left(T{\displaystyle \frac{\partial y}{\partial z}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(A_e{P}_e{\displaystyle \frac{\partial y}{\partial z}}\right)+f=0$

In the equations, $S$ represents the circumference inside the conveying pipeline, $m_r$ is the mass per unit length of the conveying pipeline, $g$ is the acceleration due to gravity, $A_e$ is the cross-sectional area outside the conveying pipeline, $A_e{P}_e$ is the pressure acting on this outer cross-section, ${\rho }_e$ is the density of the fluid outside the conveying pipeline, ${\rho }_{e}g{A}_{e}dz$ represents the buoyancy acting on the outer cross-section, $C$ is the damping coefficient, shear force $Q=-EI{\partial }^{3}y/\partial z^3$, where $EI$ is the bending stiffness of the conveying pipeline, $P_e$ is the static water pressure, and $f$ is the external force acting on the Y-direction conveying pipeline.

The force balance equation for the fluid element in Figure 2b is as follows, where Equation (8) represents the Z-direction balance equation, and Equation (9) represents the Y-direction balance equation:

(8)$-{\displaystyle \frac{\partial \left(A_i{P}_i\right)}{\partial z}}-qS-m_{f}g-m_f{\displaystyle \frac{\partial V}{\partial t}}+F{\displaystyle \frac{\partial y}{\partial z}}=0$

(9)$\begin{array}{c}{m}_f{\left({\displaystyle \frac{\partial }{\partial t}}+V{\displaystyle \frac{\partial }{\partial z}}\right)}^{2}y-F-qS{\displaystyle \frac{\partial y}{\partial z}}-{\displaystyle \frac{\partial }{\partial z}}\left(A_i{p}_i{\displaystyle \frac{\partial y}{\partial z}}\right)=0\end{array}$

In the equations, $A_i$ represents the cross-sectional area inside the conveying pipeline, $P_i$ is the internal fluid pressure, and $m_f$ is the mass per unit length of the internal fluid in the conveying pipeline.

Combining Equations (6) and (8), the following expression is derived:

(10)${\displaystyle \frac{\partial \left(T-A_i{p}_i+A_e{p}_e\right)}{\partial z}}=\left(m_r+m_f-{\rho }_e{A}_e\right)g+m_f{\displaystyle \frac{\partial V}{\partial t}}$

In the equation, $V$ represents the fluid velocity inside the conveying pipeline. Assuming $L$ is the total length of the conveying pipeline, integrating Equation (10) from $z$ to $L$ yields the following expression:

(11)$T-A_{\mathrm{i}}{p}_{\mathrm{i}}+A_{\mathrm{e}}{p}_{\mathrm{e}}=T^{\mathrm{top}}-A_{\mathrm{i}}{p}^{\mathrm{top}}(1-2\nu )-\left[\left(m_{\mathrm{r}}+m_{\mathrm{f}}-{\rho }_{\mathrm{e}}{A}_{\mathrm{e}}\right)g+m_{\mathrm{f}}{\displaystyle \frac{\partial V}{\partial t}}\right](L-z)$

In the equation, $T^{\mathrm{top}}$ is the tension at the top of the conveying pipeline, and $p^{\mathrm{top}}$ is the internal fluid pressure of the conveying pipeline. From Equations (7), (9) and (11), the y-direction and x-direction motion equations of the conveying pipeline can be obtained as follows:

(12)$EI{\displaystyle \frac{{\partial }^{4}y}{\partial z^4}}-{\displaystyle \frac{{\partial }^{2}y}{\partial z^2}}\left(T+A_e{P}_e-A_i{P}_i\right)+C{\displaystyle \frac{\partial y}{\partial t}}+m_r{\displaystyle \frac{{\partial }^{2}y}{\partial t^2}}-m_f{\left({\displaystyle \frac{\partial }{\partial t}}+V{\displaystyle \frac{\partial }{\partial z}}\right)}^{2}y=F_y$

(13)$EI{\displaystyle \frac{{\partial }^{4}x}{\partial z^4}}-{\displaystyle \frac{{\partial }^{2}x}{\partial z^2}}\left(T+A_e{P}_e-A_i{P}_i\right)+C{\displaystyle \frac{\partial x}{\partial t}}+m_r{\displaystyle \frac{{\partial }^{2}x}{\partial t^2}}-m_f{\left({\displaystyle \frac{\partial }{\partial t}}+V{\displaystyle \frac{\partial }{\partial z}}\right)}^{2}x=F_x$

In the equation,$F_x$ and $F_y$ are the forces exerted on the conveying pipeline in the x and y directions by the fluid.

3. Deep Sea Mining Pipeline Mechanical Simulation

3.1. Establishment of Deep Sea Mining System Simulation Model

In this paper, static and dynamic theoretical analyses were conducted on a deep-sea mining transportation pipeline, resulting in the motion control equations of the pipeline. The motion equation is a nonlinear partial differential equation that requires discrete numerical solutions through numerical methods. The numerical method employed in this paper is the concentrated mass method [40], where the conveying pipeline is discretized into a numerical model composed of multiple pipeline units. Mechanical and motion characteristic parameters are calculated using OrcaFlex 11.0 commercial software for simulation analysis platforms as shown in Figure 3a,b.

To study the changes in the mechanical parameters of deep-sea mining pipelines, it is necessary to consider the system as a whole, including the mechanical characteristics of both the hard lift pipe and the flexible hose under the combined effects of wind, waves, and currents. This provides simulated sensor data for constructing a model to predict pipeline mechanical properties. Thus, within OrcaFlex 11.0, dynamic models for mining vessels, subsea lifting hard pipes, lifting pumps, intermediate storage units, flexible hoses, and mining vehicles are established. The system structure is simplified as shown in Figure 4, wherein the simulation analysis utilizes a single-arch deep-sea mining system model. The blue grid represents the sea surface, and the brown grid represents the sea bed. Models for the mining hard pipe and the flexible hose are constructed based on real dimensions and material parameters. Given that the scale of the lifting pump and intermediate storage is less than one percent of the hard lift pipe, they are simplified to pipeline structures with identical mechanical properties, represented by the blue and red modules beneath the water surface in the model. For set boundary conditions, the flexible hose is fixedly connected at both ends between the intermediate storage and the mining vehicle, with the top of the hard lift pipe sharing the six degrees of freedom of the mining vessel. Models for the mining vessel and vehicle are created using the Vessel module, while the pipeline system is simulated using the Line module. The flexible hose operates at water depths of 1600 $\mathrm{m}$–1700 $\mathrm{m}$, with the buoyancy bodies forming the arch of the flexible hose simulated by the Clump module.

The relevant parameters of the hard lift pipe and the flexible hose are shown in Table 1 and Table 2, buoyancy body parameters are presented in Table 3, and the values of hydrodynamic coefficients are indicated in Table 4. This article employs the JONSWAP wave spectrum as the random wave energy spectrum for simulation analysis.

In the simulation analysis, marine environmental parameters are set based on the actual sea conditions of the target area. Specifically, wind speeds are between 1 and 3 m/s, and waves are modeled using the JONSWAP random wave spectrum. The direction of the ocean currents is shown in Figure 5, with eight primary current direction angles considered. The maximum deep-sea current is 0.1 m/s, while the surface currents are 0.2 m/s. Significant wave heights range from 0.5 to 3 m, with wave periods between 8 and 10 s. The direction of the waves is set to match the direction of the ocean currents. The density of seawater is 1035 kg/m³, and the depth of the rigid seabed is 1700 m. The velocities of the ocean currents for four return periods are presented in Table 5.

3.2. Deep-Sea Mining Conveyance Pipeline Static Simulation

The effective tension along the length of the lifting hard pipe, as obtained from static analysis, is illustrated in Figure 6. The maximum effective tension occurs at the top of the hard pipe, with a maximum value of 1509.38 kN, which is less than the maximum allowable tension of 10,000 kN for the lifting hard pipe. The effective tension the lifting hard pipe experiences decreases gradually along its length. Due to the top end of the hard pipe being hinged to the mining vessel, the positions of the hard pipe closer to the top are more significantly affected by the heave motion of the mining vessel and wave forces.

The effective tension along the length of the flexible hose is shown in Figure 7. At 0 $\mathrm{m}$ and 300 m lengths of the hose, it is connected to the mining vehicle and the intermediate storage, respectively. Between the positions of 100 m–150 m along the hose, the effective tension is less than at the two peak points. This reduction in tension is due to the placement of buoyancy bodies within this interval, which decreases the effective tension exerted on the pipeline under the influence of buoyancy. The maximum effective tension distributed along the length of the hose is 26.87 kN, occurring at 107.5 m, which is less than the maximum allowable tension of 200 kN for the flexible hose.

The distribution of the bending radius along the length of the flexible hose is shown in Figure 8. The minimum bending radius value distributed along the hose length is 5.13 m, located at 132.5 m of the hose, which is greater than the minimum allowable bending radius of 1 m for the flexible hose.

The bending moment is a parameter related to the characteristics of the bending radius. By ensuring that the minimum value of the bending radius distributed along the hose length falls within the normal working requirements and conducting a static analysis of the bending moment, a more comprehensive study of the mechanical properties of the deep-sea mining conveyance pipeline can be achieved. As shown in Figure 9, the maximum value of the bending moment distributed along the hose length is 2.66 kNm, located at 132.5 m of the hose. The position of the minimum value of the bending radius along the hose length is also the position with the maximum value of the bending moment distributed along the hose length.

From the static analysis, it is known that when the deep-sea mining conveyance pipeline is in static equilibrium, both the maximum effective tension distributed along the pipeline and the minimum value of the bending radius distributed along the hose fall within the design performance requirements of the pipeline. The characteristics of the bending moment are closely related to the bending radius. In subsequent dynamic analyses, these points that represent the dangerous spots in terms of the pipeline’s force-bearing and bending characteristics will be selected for multiple working condition dynamic analyses. Simulation sensor data will serve as the basis for constructing a predictive model of mechanical properties.

3.3. Orthogonal Experimental Design and Dynamic Simulation

Due to the influence of various ocean environmental parameters on the deep-sea mining system, a large number of tests are required to comprehensively simulate the load-bearing conditions of the deep-sea mining conveyance pipeline under various working conditions. The orthogonal experimental design can effectively combine the levels of each parameter, providing a comprehensive assessment of the effects of different factor combinations, reducing the number of repetitive tests, and improving test efficiency. In this study, the orthogonal experimental design scheme was formulated using the SPSS statistical tool [6], combining the working conditions within the range of ocean environmental parameters introduced earlier, as shown in Table 6. The wind speed, wave period, and significant wave height were each set at intervals of 0.5. As the deep-sea mining conveyance pipeline operates at different depths, it is significantly influenced by variations in ocean current factors. Thus, the study considered multiple ocean current speeds in the region where the deep-sea mining operation is conducted, including one common condition and four recurrence period conditions. The ocean current angles were set to the most frequent levels, focusing on the main eight ocean current angles introduced earlier. The experiment included five factors: wind, waves, and currents. Each factor was assigned 5, 5, 6, 5, and 8 levels, respectively.

If a full factorial experiment were conducted, it would require completing 6000 sets of experiments, necessitating extensive computational resources and time. By utilizing an orthogonal experimental design, an L₆₄ orthogonal array was selected to accommodate 5 factors and 8 levels as shown in Table 7, resulting in a total design of 64 combinations of working conditions. The ocean current speed is represented by the ratio of the surface current speed to the deep current speed. Through dynamic simulation calculations, the pipeline’s mechanical response parameters were determined. The pipeline strain, effective tension, bending radius, and bending moment were selected as the outcomes of the orthogonal experiment.

Obtaining a substantial amount of simulation sensor data through dynamic simulation is essential for constructing a predictive model of pipeline mechanical properties. The volume of data significantly impacts the model’s prediction error, where algorithms trained on less data tend to have larger errors. Thus, based on experience, this study set tens of thousands of data sets for training the neural network model. Dynamic simulations were conducted for the 64 sets of working conditions derived from the orthogonal experiment, with each simulation set to 50 s and a time step of 0.1 s, resulting in 501 data points per simulation. Consequently, the experiment generated a total of 32,064 sets of simulation sensor data for training and testing the pipeline mechanical properties prediction model. Given the vast amount of data, this paper selects and displays part of the simulation time history data from the first set of experimental conditions for demonstration. When the wind speed is 2.5 m/s, the wave period is 10 s, the significant wave height is 1 m, and the surface current speed is 2 m/s, the time history data of mechanical parameters for both the lifting hard pipe and the flexible hose are presented in Table 8 and Table 9, respectively. The mechanical properties of the pipeline exhibit slight variations over time.

4. Construction of the Predictive Model for Pipeline Mechanical Characteristics

4.1. Construction Method of Pipeline Mechanical Properties Prediction Model

4.1.1. Integrated Simulation Platform

To build a predictive model, substantial and comprehensive data are required. This paper uses the dynamic simulation results of multiple working conditions from orthogonal experiment combinations as simulation sensor data for training the machine learning algorithm model. An online simulation platform that enables interaction between OrcaFlex 11.0 and Matlab R2022a has been established. The interaction flow between the platforms is illustrated in Figure 10. Upon completion of the simulation analysis model, a connection is made to Matlab through the OrcFxAPI module of OrcaFlex, where the setting and execution of parameters for multiple simulation analysis conditions in OrcaFlex are completed in Matlab. OrcaFlex performs static and dynamic analyses of the system, with the simulation sensor data outputs exported to Matlab to complete the training and testing of the machine learning algorithm model, thus constructing a prediction model with relatively high accuracy.

4.1.2. Basic Structure of BP Neural Network

The BP neural network is employed to train simulation sensor data and build a prediction model [41]. The neural network structure is depicted in Figure 11. The neural network outcomes consist of multiple neurons interconnected through weights and thresholds between input, hidden, and output neurons. The expression for the final output layer is as follows:

(14)$y_k=f(\sum _{j=1}^n{W}_{jk}f(\sum _{i=1}^m{W}_{ij}{x}_i-b_j)-b_k)$

In the equation, where $y_k$ represents the output parameter and $x_i$ denotes the input training sample, the expression for the activation function $f\left(x\right)$ is defined as $f\left(x\right)={\displaystyle \frac{1}{1+e^{-x}}}$. $W_{ij}$ and $W_{jk}$ represent the weights between hidden parameters and input or output parameters, while $b_j$ and $b_k$ are the thresholds for hidden and output parameters. Here, $m$ stands for the number of input parameters, $k$ represents the number of output parameters, and $n$ signifies the number of hidden parameters, where $n=\sqrt{m+k}+a$. The value of $a$ ranges from 1 to 10 [42].

4.1.3. Evaluation Metrics

Based on the results of the trained BP neural network model, the effectiveness of the model training is typically assessed through metrics such as the mean absolute error (MAE), root mean square error (RMSE), and correlation coefficient [43]. The calculation formulas are as follows:

Mean absolute error (MAE):

(15)$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{h}}{\sum }_{i=1}^h|Y_i-{\widehat{Y}}_i|$

Root mean square error (RMSE):

(16)$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{h}}{\sum }_{i=1}^h{\left(Y_i-{\widehat{Y}}_i\right)}^2}$

Correlation coefficient:

(17)$\mathrm{R}={\displaystyle \frac{\sum _{i=1}^h(Y_i-\bar{Y})({\widehat{Y}}_i-{\widehat{Y}}_i^{\prime })}{\sqrt{\sum _{i=1}^h(Y_i-\bar{Y}{)}^2\sum _{i=1}^h({\widehat{Y}}_i-{\widehat{Y}}_i^{\prime }{)}^2}}}$

In the above evaluation metric calculation formulas, $h$ represents the sample size, $Y_i$ denotes the original sample values, $\bar{Y}$ is the sample mean value, ${\widehat{Y}}_i$ stands for the predicted values, and ${\widehat{Y}}_i^{\prime }$ represents the mean of the predicted values.

4.2. Model for Predicting the Mechanical Properties of Rigid Lifting Pipe

According to the static analysis, the maximum effective tension along the length of the rigid lifting pipe is located at the top end of the pipe and reaches a value of 1509.38 kN. During the dynamic simulation process, the maximum effective tension distribution along the length of the rigid lifting pipe remains at the top end. To ensure the operational safety of the deep-sea mining system, it is imperative to monitor the real-time changes in the effective tension at the critical stress points along the length of the pipe. Therefore, this study uses the 32,064 sets of data obtained from 64 groups of simulations designed through orthogonal experiments to train a BP neural network model.

Based on the BP neural network structure setup, the simulation sensor data, which includes sea wind speed, wave period, significant wave height, ocean current speed, ocean current angle, and pipeline strain, are used as the six input layers of the BP neural network. The effective tension at the top end of the rigid lifting pipe serves as the output layer. The neural network topology is illustrated in Figure 12.

In this study, 80% of the data set (i.e., 25,651 data sets) is selected as the training set for the BP neural network, while the remaining 20% (i.e., 6413 data sets) are used as the test set. The training process is executed 1000 times with a learning rate of 0.01, and the training objective is to achieve a minimum error of 0.00001. The digital twin model is constructed by training the neural network using simulation sensor data.

The performance of the BP neural network model is evaluated using the mean absolute error (MAE), root mean square error (RMSE), and correlation coefficient. The closer the RMSE is to 0, the better the training effect of the model. A correlation coefficient closer to 1 indicates a strong positive linear correlation between the predicted values and the simulation sensor data. The prediction results of the effective tension at the top end of the rigid lifting pipe using the BP neural network model are shown in Figure 13. As can be seen from the figure, the effective tension at the top end of the rigid lifting pipe varies under different sea conditions.

The results in Figure 13 indicate that the neural network model accurately predicts the effective tension at the top end of the rigid pipe. The red predicted curve closely aligns with the blue simulated value curve in most of the data regions. The mean absolute error (MAE) is 0.2968, the root mean square error (RMSE) is 0.8552, and the correlation coefficient is 0.9963. The maximum value of the effective tension at the top end of the rigid lifting pipe is 1689.27 kN, which is less than the maximum allowable tension of 10,000 kN for the rigid lifting pipe. Therefore, in the 64 operational scenarios based on the actual sea conditions, the maximum effective tension experienced by the deep-sea mining rigid lifting pipe is significantly lower than the maximum allowable effective tension for the pipe. This neural network model can accurately predict the effective tension at the top end of the rigid lifting pipe, forming a predictive model for determining the maximum effective tension distribution along the length of the rigid lifting pipe under different operational conditions.

4.3. Model for Predicting the Mechanical Properties of Flexible Hose

The model for predicting the mechanical properties of the flexible hose is based on static analysis, where the maximum effective tension experienced by the hose is located at a length of 107.5 m along the hose. The minimum bending radius and maximum bending moment along the length of the hose occur at a length of 132.5 m. During the dynamic simulation process, the maximum effective tension, minimum bending radius, and maximum bending moment along the length of the hose remain consistent, with these critical points identified through static analysis, providing the mechanical parameters for dynamic analysis. This ensures the reliability of studying the hose’s behavior under dynamic conditions. Similar to the construction of the tension prediction model for the rigid lifting pipe in Section 4.2, this study utilizes 64 sets of simulations designed through orthogonal experiments, resulting in 32,064 data sets used for training the BP neural network model. The simulation sensor data, including sea wind speed, wave period, significant wave height, ocean current speed, ocean current angle, and pipeline strain, are used as the six input layers of the BP neural network. The maximum effective tension, minimum bending radius, and maximum bending moment experienced by the hose are set as the output layers, with a neural network topology similar to that shown in Figure 13.

The results of the training of the effective tension prediction model at a length of 107.5 m along the hose are depicted in Figure 14. The predicted effective tension at this point aligns closely with the simulated effective tension values, with an average absolute error (MAE) of 0.17854, a root mean square error (RMSE) of 0.2069, and a correlation coefficient of 0.9872. These metrics demonstrate the model’s good reliability in accurately predicting the maximum effective tension of the hose. The maximum effective tension at a length of 107.5 m along the hose is 29.38 kN, which is below the maximum allowable effective tension of 200 kN for the hose. In all 64 operational scenarios, the hose operates within a safe tension range.

The model for predicting the bending radius at a length of 132.5 m along the hose is shown in Figure 15. The predicted bending radius at this point closely aligns with the simulated bending radius values. The prediction model has an average absolute error (MAE) of 0.1436, a root mean square error (RMSE) of 0.1648, and a correlation coefficient of 0.9695, indicating that the model is reliable and can accurately predict the minimum bending radius of the hose. The minimum value of the bending radius at a length of 132.5 m along the hose is 6.12 m, which is greater than the minimum allowable bending radius of 1 m for the hose. This implies that the hose will not undergo excessive bending deformation during operations in all 64 operational scenarios.

The training results of the model for predicting the bending moment at a length of 132.5 m along the hose are illustrated in Figure 16. The predicted bending moment at this point closely aligns with the simulated bending moment values, with an average absolute error (MAE) of 0.0367, a root mean square error (RMSE) of 0.0530, and a correlation coefficient of 0.9725. These metrics indicate that the model is reliable and can accurately predict the maximum bending moment of the hose. The maximum value of the bending moment at a length of 132.5 m along the hose is 3.43 kNm.

The aforementioned BP neural network models can accurately predict the mechanical parameters of the hose based on the sensor-detectable ocean environmental parameters and simulation sensor data, thereby establishing a predictive model for the mechanical characteristics of deep-sea mining transportation hoses. The results from the 64 simulations indicate that the stress and bending deformation of the hose vary under different sea conditions. Therefore, the accurate monitoring of the mechanical state parameters of the hose through the predictive model is necessary to ensure the operational safety of deep-sea mining systems.

Among the same wind speed and wave parameters, the ocean current speed and ocean current angle have a more significant impact on deep-sea mining pipelines. The variation patterns of the effective tension at a length of 107.5 m along the hose, the minimum bending radius at a length of 132.5 m, and the bending moment are illustrated in Figure 17a, Figure 17b, and Figure 17c, respectively.

Due to the changes in the shape of the hose during deep-sea mining operations with variations in operational conditions and sea conditions, the position of the maximum effective tension, maximum bending moment, and minimum bending radius along the length of the hose may vary. Therefore, this study constructs a predictive model for the mechanical parameters of each node of the hose. The simulated model of the hose consists of 102 nodes, and the prediction model for the overall mechanical characteristics of the hose involves predicting the mechanical parameters of each node. The construction method of the predictive model for the overall mechanical characteristics of the hose is similar to the construction of the digital twin model for the critical points mentioned earlier. The BP neural network model’s predictive effectiveness for the mechanical parameters of each node of the hose is illustrated in Figure 18.

The results in Figure 18a indicate that the average absolute error (MAE) values for the effective tension prediction of each node of the hose range between 0.0587 and 0.8328, for the prediction of each node’s bending radius, the MAE values range between 0.0576 and 0.6473, and for the prediction of each node’s bending moment, the MAE values range from 0.0171 to 0.2305. The root mean square error (RMSE) trends for model training are consistent with the MAE values, with the RMSE values for the effective tension prediction of each node ranging between 0.0816 and 1.0116, for the prediction of each node’s bending radius, the RMSE values range between 0.0890 and 0.8330, and for the bending moment prediction of each node, the RMSE values range from 0.0238 to 0.3244. The results from the model training indicate that the BP neural network model performs well in predicting the mechanical parameters of each node of the hose, with the RMSE values close to 0, demonstrating a certain level of reliability in the model’s training effect. As shown in Figure 18b, the correlation coefficients for the effective tension prediction of each node range between 0.9772 and 0.9854, for the bending radius prediction of each node, the correlation coefficients range between 0.9656 and 0.9819, and for the bending moment prediction of each node, the correlation coefficients range from 0.9711 to 0.9907. With correlation coefficients close to 1, indicating a linear relationship between predicted and simulated values, the model exhibits good prediction performance.

5. Conclusions

Based on the analysis of the mechanical theory of deep-sea mining transportation pipelines, this paper establishes a simulation model for deep-sea mining systems and conducts static and dynamic analyses of the deep-sea mining transportation pipelines. It innovatively combines 64 sets of working condition test schemes through orthogonal experiments and establishes, for the first time, a pipeline mechanical characteristic prediction model based on the BP neural network. The main work and conclusions of this paper are as follows:

• (1). In the OrcaFlex 11.0 software, an overall interactive simulation analysis model of the deep-sea mining system is established. A static analysis of the lifting hard pipe and hose is conducted to obtain key parameters of the pipeline’s mechanical characteristics. Using an orthogonal experimental design, a total of 64 sets of experimental conditions are combined to analyze the dynamic simulation of the deep-sea mining transportation pipeline system. A total of 32,064 sets of simulated sensor data are obtained for training and testing the mechanical prediction model of the pipeline.

• (2). A predictive model is constructed based on the BP neural network for the effective tension at the top of the lifting hard pipe, the effective tension at the position of 107.5 m along the hose, the bending radius at the position of 132.5 m along the hose, and the bending moment. The impact of ocean current speed and angle on the mechanical parameters of the hose is studied, and the mechanical parameters of each node of the hose are predicted. The trained BP neural network model has average absolute error (MAE) and root mean square error (RMSE) values close to 0, with correlation coefficients close to 1. There is a strong linear positive correlation between the predicted values and the simulated values. The trained neural network model can serve as a predictive model for the mechanical characteristics of the transportation pipeline. By inputting sensor data, it can predict the changes in the mechanical parameters of critical points in the transportation pipeline, ensuring that the tension of the lifting hard pipe, hose tension, bending radius of the hose, and bending moment remain within safe limits. In the event of safety risks, adjustments to the mining operation plan can be made promptly to avoid pipeline failure, breakage, or excessive bending deformation.

Currently, in actual deep-sea mining operations, the sensor parameters related to the mechanical properties of the conveying pipelines that can be monitored include wind speed, wave period, significant wave height, current speed, current angle, and the strain of the pipeline. This paper innovatively takes the dynamic simulation results of multiple working conditions from the earlier orthogonal experiment combinations as simulation sensor data. By employing machine learning methods, a predictive model of pipeline mechanical properties is constructed for the first time, with the monitorable sensor parameters serving as input for model training. This model predicts mechanical parameters that are difficult to monitor in actual deep-sea mining operations. It analyzes the variation patterns of mechanical parameters of the conveying pipeline under different environmental loads to ensure that the pipeline operates within the mechanical parameter range designed for the pipeline. This study can ensure the safety of deep-sea mining operations by predicting the performance of pipelines in extreme marine environments, identifying and addressing potential structural weaknesses in advance, and helping to extend the service life of pipeline systems. In addition, predicting mechanical properties can promote an efficient exploitation of deep-sea resources and support the goal of sustainable development.

Author Contributions

Conceptualization, Q.H. and Y.Q.; methodology, Y.Q.; software, J.Z. and M.Z.; validation, Q.H., J.H. and M.Z.; formal analysis, Y.Q.; investigation, Y.Q., J.Z. and J.H.; resources, Q.H.; data curation, Y.Q.; writing—original draft preparation, Y.Q.; writing—review and editing, Y.Q., J.Z. and Q.H.; visualization, Y.O.; supervision, Q.H.; project administration, Q.H.; funding acquisition, Q.H. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy restrictions.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive suggestions, which comprehensively improved the quality of the paper. All individuals have consented to the acknowledgement.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Static analysis of pipeline unit catenary line.

Figure 2. Force analysis of micro-elements of deep-sea mining transportation pipelines: (a) force analysis on the pipeline wall; (b) fluid force analysis inside the pipeline.

Figure 3. Orcaflex simulation analysis platform: (a) simulation scenario near the sea surface; (b) simulation scenario near the seabed.

Figure 4. Schematic diagram of deep-sea mining system simulation model.

Figure 5. Top view of deep-sea mining system and ocean current direction.

Figure 6. Distribution of effective tension along the length of the lifting hard pipe.

Figure 7. Distribution of effective tension along the length of the flexible hose.

Figure 8. Distribution of bending radius along the length of the flexible hose.

Figure 9. Distribution of bending moment along the length of the flexible hose.

Figure 10. Online simulation platform with OrcaFlex and Matlab interaction.

Figure 11. Fundamental structure of the BP neural network model for pipeline mechanical characteristics.

Figure 12. Topological structure of the BP neural network for pipeline mechanical characteristics.

Figure 13. Prediction of effective tension at the top end of the mining hard pipe.

Figure 14. Prediction of effective tension at flexible hose length of 107.5 m.

Figure 15. Prediction of bending radius at flexible hose length of 132.5 m.

Figure 16. Prediction of bending moment at flexible hose length of 132.5 m.

Figure 17. Influence of ocean current speed and angle on the mechanical parameters of flexible hose: (a) variation in effective tension at flexible hose length of 107.5 m; (b) variation in bending radius at flexible hose length of 132.5 m; (c) variation in bending moment at flexible hose length of 132.5 m.

Figure 18. Prediction effectiveness of mechanical parameters for each node on the flexible hose by the BP neural network model: (a) MAE value; (b) correlation coefficient.

Table 1

Mining riser parameters.

Pipe Performance	Parameter Value
Inner diameter (mm)	224
Outer diameter (mm)	244
Length (m)	1619
Weight in air (kg/m)	38.4
Weight in water (kg/m)	56.9
Yield strength (MPa)	758 (${\sigma }_{\mathrm{bmin}}$), 965 (${\sigma }_{\mathrm{bmax}}$)
Maximum allowable effective tension (kN)	10000

Table 2

Hose parameters.

Pipe Performance	Parameter Value
Inner diameter (mm)	200
Outer diameter (mm)	265
Length (m)	300
Weight in water (kg/m)	16.9
Tensile stiffness (kN/m)	732
Bending stiffness (kN·m2)	3.33
Maximum allowable effective tension (kN)	200
Minimum allowable bending radius (m)	1
Compressive strength (MPa)	1.9
Bulk modulus (MPa)	101.7

Table 3

Buoyancy module parameters.

Parameter Type	Parameter Value
Inner diameter (mm)	0.26
Outer diameter (mm)	0.94
Weight per buoyancy module (kg)	375
Buoyancy provided per module (kg)	334

Table 4

Hydrodynamic coefficients.

Coefficient Type	Parameter Value
Horizontal drag coefficient	1.2
Horizontal inertia coefficient	1.5
Axial drag coefficient	0.024
Axial inertia coefficient	1.0

Table 5

Ocean current speeds for various return periods.

Water Depth (m)	Ocean Current Speed (m/s)
	1-Year Return	10-Year Return	50-Year Return	100-Year Return
0.00	0.86	0.95	1.01	1.03
30.38	0.82	0.91	0.97	0.99
299.80	0.61	0.66	0.69	0.71
499.59	0.39	0.41	0.42	0.42
699.50	0.38	0.39	0.40	0.41
899.28	0.36	0.38	0.39	0.39
1099.07	0.36	0.37	0.38	0.39
1298.98	0.35	0.37	0.38	0.38
1498.77	0.35	0.37	0.38	0.38
1698.67	0.35	0.37	0.38	0.38
1712.50	0.23	0.25	0.26	0.26

Table 6

Orthogonal experimental factors and levels.

Level	Wind Speed (m/s)	Wave Parameters		Ocean Current Parameters
		Wave Period (s)	Significant Wave Height (m)	Surface/Deep Flow (m/s)	Ocean Current Angles (deg)
1	1	8	0.5	0.2/0.1	0
2	1.5	8.5	1	0.86/0.23	45
3	2	9	1.5	0.95/0.25	90
4	2.5	9.5	2	1.01/0.26	135
5	3	10	2.5	1.03/0.26	180
6	-	-	3	-	225
7	-	-	-	-	270
8	-	-	-	-	315

Table 7

Orthogonal experimental design (Due to space limitations, these rows with dots have omitted the data in the middle, the same for the following tables).

Experiment Number	Wind Speed (m/s)	Wave Period (s)	Significant Wave Height (m)	Ocean Current Speed (m/s)	Ocean Current Angle (deg)
1	2.5	10	1	0.2	270
2	1.5	9	3	0.2	270
3	2.5	8.5	2.5	0.2	135
…	…	…	…	…	…
35	1.5	8	1.5	0.95	45
36	1.5	9.5	0.5	0.95	270
37	1	9	1.5	0.95	90
…	…	…	…	…	…
62	1.5	8.5	1	1.03	90
63	3	9.5	1.5	1.03	0
64	1	10	2.5	1.03	180

Table 8

Time history data for mechanical parameters of mining rigid pipe in test condition 1.

Simulation Time (s)	Lifting Hard Pipe Top Strain (%)	Lifting Hard Pipe Top Effective Tension (kN)
0.1	0.2019	1478.5228
0.2	0.2017	1476.5988
…	…	…
25.0	0.2016	1476.2897
25.1	0.2019	1478.6173
…	…	…
49.9	0.2027	1484.0249
50.0	0.2020	1479.2711

Table 9

Time history data for mechanical parameters of flexible hose in test condition 1.

Simulation Time (s)	Strain at 107.5 m (%)	Effective Tension at 107.5 m (kN)	Strain at 132.5 m (%)	Bending Radius at 132.5 m (m)	Bending Moment at 132.5 m (kNm)
0.1	0.0341	27.5893	0.0341	7.5405	2.7796
0.2	0.0333	27.5856	0.0339	7.5388	2.7802
…	…	…	…	…	…
25.0	0.0332	27.5181	0.0340	7.5317	2.7828
25.1	0.0331	27.5166	0.0341	7.5318	2.7826
…	…	…	…	…	…
49.9	0.0333	27.5983	0.0341	7.5610	2.7721
50.0	0.0334	27.5984	0.0343	7.5615	2.7720

References

1. Watzel, R.; Rühlemann, C.; Vink, A. Mining mineral resources from the seabed: Opportunities and challenges. Mar. Policy; 2020; 114, 103828. [DOI: https://dx.doi.org/10.1016/j.marpol.2020.103828]

2. Petersen, S.; Kraeschell, A.; Augustin, N.; Jamieson, J.; Hein, J.R.; Hannington, M.D. News from the seabed—Geological characteristics and resource potential of deep-sea mineral resources. Mar. Policy; 2016; 70, pp. 175-187. [DOI: https://dx.doi.org/10.1016/j.marpol.2016.03.012]

3. Hu, Q.; Li, Z.; Zhai, X.; Zheng, H. Development of Hydraulic Lifting System of Deep-Sea Mineral Resources. Minerals; 2022; 12, 1319. [DOI: https://dx.doi.org/10.3390/min12101319]

4. Wang, G.; Liu, S.-j.; Li, L. FEM modeling for 3D dynamic analysis of deep-ocean mining pipeline and its experimental verification. J. Cent. South Univ. Technol.; 2007; 14, pp. 808-813. [DOI: https://dx.doi.org/10.1007/s11771-007-0154-5]

5. Chen, J.J.; Wang, D.J. Nonlinear Dynamic Analysis of Flexible Marine-Risers. China Ocean. Eng.; 1991; 5, pp. 373-384.

6. Mavrakos, S.A.; Chatjigeorgiou, I.K. Assessment of bottom-cable interaction effects on mooring line dynamics. Proceedings of the 17th International Conference on Offshore Mechanics and Arctic Engineering (OMAE’ 98); Lisbon, Portugal, 5–9 July 1998.

7. Vogel, H.; Natvig, B.J. Dynamics of Flexible Hose Riser Systems. J. Offshore Mech. Arct. Eng.; 1987; 109, pp. 244-248. [DOI: https://dx.doi.org/10.1115/1.3257016]

8. Felippa, C.A.; Chung, J.S. Nonlinear Static Analysis of Deep Ocean Mining Pipe—Part I: Modeling and Formulation. J. Energy Resour. Technol.; 1981; 103, pp. 11-15. [DOI: https://dx.doi.org/10.1115/1.3230807]

9. Chung, J.S.; Felippa, C.A. Nonlinear Static Analysis of Deep Ocean Mining Pipe—Part II: Numerical Studies. J. Energy Resour. Technol.; 1981; 103, 16. [DOI: https://dx.doi.org/10.1115/1.3230808]

10. Chung, J.S.; Whitney, A.K.; Loden, W.A. Nonlinear Transient Motion Of Deep Ocean Mining Pipe. J. Energy Resour. Technol.; 1981; 103, 2. [DOI: https://dx.doi.org/10.4043/3832-MS]

11. Chung, J.S.; Whitney, A.K. Dynamic Vertical Stretching Oscillation of an 18,000-Ft Ocean Mining Pipe. Proceedings of the Offshore Technology Conference; Houston, TX, USA, 4–7 May 1981; OTC: Boca Raton, FL, USA, 1981.

12. Brink, A.N.; Chung, J.S. Automatic Position Control of a 30,000 Tons Ship During Ocean Mining Operations. Proceedings of the Offshore Technology Conference; Houston, TX, USA, 4–7 May 1981; OTC: Boca Raton, FL, USA, 1981; [DOI: https://dx.doi.org/10.4043/4091-MS]

13. Chung, J.S.; Cheng, B.R.; Huttelmaier, H.P. Three-dimensional coupled responses of a vertical deep-ocean pipe: Part II. Excitation at pipe top and external torsion. Int. J. Offshore Polar Eng.; 1994; 4, pp. 320-330.

14. Chung, J.S. Track-keeping control of seafloor miner by successive learning of unknown velocity and soil properties. Proceedings of the ISOPE Ocean Mining and Gas Hydrates Symposium; Goa, India, 8–10 November 1999; ISOPE: Mountain View, CA, USA, 1999.

15. Chung, J.S. Deep-ocean mining technology: Learning curve I. Proceedings of the ISOPE Ocean Mining and Gas Hydrates Symposium; Tsukuba, Japan, 15–19 September 2003; ISOPE: Mountain View, CA, USA, 2003.

16. Cheng, B.R.; Zheng, Z.C.; Chung, J.S. Effects of flexible joints on the 3-D nonlinear coupled responses of a long vertical pipe. Proceedings of the ISOPE 5th International Offshore and Polar Engineering Conference; The Hague, The Netherlands, 11–16 June 1995; ISOPE: Mountain View, CA, USA, 1995.

17. Cheng, B.; Chung, J.S. Effects of Axial Dampers And Elastic Joints On the 3-D Dynamic Responses of a Deep-Ocean Pipe With Torsional Coupling. Int. J. Offshore Polar Eng.; 1997; 7, pp. 36-43.

18. Chung, J.S.; Cheng, B.S. MSE And FEM Modeling of Thrusts to Elastic Joints of Long Vertical Pipe In 3-D Nonlinear Motions. Int. J. Offshore Polar Eng.; 1999; 9, pp. 83-96.

19. Chung, J.S.; Cheng, B.R. Effects of Elastic Joints on 3-D Nonlinear Responses of a Deep-Ocean Pipe: Modeling and Boundary Conditions. Int. J. Offshore Polar Eng.; 1996; 6, pp. 203-211.

20. Cheng, B.; Chung, J.S.; Isope, O.P.E. Application of Thrusts to Elastic Joints on Long Vertical Pipe in 3-D Nonlinear Motions—Part II: Numerical Examples by MSE and FEM Results. Proceedings of the ISOPE 8th International Offshore and Polar Engineering Conference; Montreal, QC, Canada, 24–29 May 1998; ISOPE: Mountain View, CA, USA, 1998.

21. Mustoe, G.G.W.; Huttelmaier, H.; Chung, J.S. Assessment of Dynamic Coupled Bending-Axial Effects For Two-Dimensional Deep-Ocean Pipes By the Discrete Element Method. Int. J. Offshore Polar Eng.; 1992; 2, pp. 289-296.

22. Mustoe, G.G.W.; Hettelmaier, H.; Chung, J.S. Dynamic Coupled Bending-Axial Analysis Of Two-Dimensional Deep-Ocean Pipes By The Discrete Element Method. Proceedings of the ISOPE 2nd International Offshore and Polar Engineering Conference; San Francisco, CA, USA, 14–19 June 1992; ISOPE: Mountain View, CA, USA, 1992.

23. Huttelmaier, H.P.; Chung, J.S.; Mustoe, G.G.W.; Zheng, Z.; Cheng, B. Eigenvalues of a Long Vertical Pipe By DEM, FEM And Exact Solution. Proceedings of the ISOPE 3rd International Offshore and Polar Engineering Conference; Singapore, 6–11 June 1993; ISOPE: Mountain View, CA, USA, 1993.

24. Hong, S.; Choi, J.S.; Kim, H.W. Effects of internal flow on dynamics of underwater flexible pipes. Proceedings of the 5th ISOPE Ocean Mining Symposium; Tsukuba, Japan, 15–19 September 2003.

25. Hong, S.; Kim, H.W. Coupled Dynamic Analyses of Underwater Tracked Vehicle and Long Flexible Pipe. Proceedings of the Sixth (2005) ISOPE Ocean Mining Symposium; Changsha, China, 10–13 October 2005; pp. 132-140.

26. Wu, Z.W.; Xie, C.R.; Mei, G.X.; Dong, H.Y. Dynamic analysis of parametrically excited marine riser under simultaneous stochastic waves and vortex. Adv. Struct. Eng.; 2019; 22, pp. 268-283. [DOI: https://dx.doi.org/10.1177/1369433218783968]

27. Ghadimi, R. A simple and efficient algorithm for the static and dynamic analysis of flexible marine risers. Comput. Struct.; 1988; 29, pp. 541-555. [DOI: https://dx.doi.org/10.1016/0045-7949(88)90364-1]

28. Ablow, C.M.; Schechter, S. Numerical simulation of undersea cable dynamics. Ocean. Eng.; 1983; 10, pp. 443-457. [DOI: https://dx.doi.org/10.1016/0029-8018(83)90046-X]

29. Bernitsas, M.M.; Kokarakis, J.E.; Imron, A. Large deformation three-dimensional static analysis of deep water marine risers. Appl. Ocean. Res.; 1985; 7, pp. 178-187. [DOI: https://dx.doi.org/10.1016/0141-1187(85)90024-0]

30. Nordgren, R.P. On Computation of the Motion of Elastic Rods. J. Appl. Mech.; 1974; 41, 13. [DOI: https://dx.doi.org/10.1115/1.3423387]

31. Xu, H.L.; Chen, W.; Li, L.; Yang, F.Q. Morphology of deep-sea mining hydraulic conveying pipeline and its influencing laws in marine dynamic environment. J. Cent. South Univ.; 2023; 30, pp. 3790-3807. [DOI: https://dx.doi.org/10.1007/s11771-022-5032-7]

32. Wang, L.Q.; Ju, M.; Xing, X.D.; Yun, F.H.; Wang, X.Y. Dynamic Behavior of the Deepwater Flexible Pipeline during Pipe Laying Process. J. Mar. Sci. Eng.; 2020; 8, 286. [DOI: https://dx.doi.org/10.3390/jmse8040286]

33. Buljak, V.; Cocchetti, G.; Cornaggia, A.; Garbowski, T.; Maier, G.; Novati, G. Materials Mechanical Characterizations and Structural Diagnoses by Inverse Analyses. Handbook of Damage Mechanics: Nano to Macro Scale for Materials and Structures; Voyiadjis, G.Z. Springer: New York, NY, USA, 2015; pp. 619-642.

34. Buljak, V.; Cocchetti, G.; Cornaggia, A.; Maier, G. Assessment of residual stresses and mechanical characterization of materials by “hole drilling” and indentation tests combined and by inverse analysis. Mech. Res. Commun.; 2015; 68, pp. 18-24. [DOI: https://dx.doi.org/10.1016/j.mechrescom.2015.02.003]

35. Zhang, Y.; Quan, S.; Liu, G. Residual Stress Evaluation of Metallic Component Based on the Synthetic Nonlinear Coefficient. Proceedings of the MEACM 2020: Mechanical Engineering and Applied Composite Materials; Beijing, China, 24–25 October 2020; Springer: Cham, Switzerland, 2021; pp. 45-52.

36. Song, L.; Wang, D.L.; Liu, X.W.; Yin, A.J.; Long, Z.D. Prediction of mechanical properties of composite materials using multimodal fusion learning. Sens. Actuator A-Phys.; 2023; 358, 13. [DOI: https://dx.doi.org/10.1016/j.sna.2023.114433]

37. Kang, Z.; Zhang, L.; Zhang, X. Analysis on J lay of SCR based on catenary and large deflection beam theory. Ocean. Eng.; 2015; 104, pp. 276-282. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2015.05.017]

38. Gong, S.F.; Xu, P. The influence of sea state on dynamic behaviour of offshore pipelines for deepwater S-lay. Ocean Eng.; 2016; 111, pp. 398-413. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2015.11.013]

39. Chaudhuri, C.H.; Choudhury, D. Buried pipeline subjected to seismic landslide: A simplified analytical solution. Soil Dyn. Earthq. Eng.; 2020; 134, 6. [DOI: https://dx.doi.org/10.1016/j.soildyn.2020.106155]

40. Di, F.D.; Zhang, C.Y.; Yin, J.; Sun, L.M.; Chen, L. Effects of Concentrated Mass on the In-Plane Dynamic Behaviors of Two-Cable Networks with a Cross-Tie. Int. J. Struct. Stab. Dyn.; 2024; 20, e2755. [DOI: https://dx.doi.org/10.1142/s0219455425500464]

41. Han, F.; Du, F.; Jiao, S.; Zou, K. Predictive Analysis of a Building’s Power Consumption Based on Digital Twin Platforms. Energies; 2024; 17, 3692. [DOI: https://dx.doi.org/10.3390/en17153692]

42. Peta, K.; Żurek, J. Prediction of air leakage in heat exchangers for automotive applications using artificial neural networks. Proceedings of the 2018 9th IEEE Annual Ubiquitous Computing, Electronics & Mobile Communication Conference (UEMCON); New York, NY, USA, 8–10 November 2018; pp. 721-725. [DOI: https://dx.doi.org/10.1109/UEMCON.2018.8796666]

43. Yang, Q.; Zhang, J.; Dai, X.; Ye, Z.; Wang, C.; Lu, S. Research on Permeability Characteristics and Gradation of Rockfill Material Based on Machine Learning. Water; 2024; 16, 1135. [DOI: https://dx.doi.org/10.3390/w16081135]