A numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows

Abstract

The present invention provides a numerical calculation method for studying the nonlinear vortex-induced vibration model of the submarine inclined pipeline under the excitation of internal and external flow coupling, and adopts GFDM to introduce and solve the high-order nonlinear differential equations of the vortex-induced vibration of the submarine pipeline under the excitation of internal and external flow. Because GFDM is only applied to nonlinear partial differential equations with the highest order being the second order, and the high-order nonlinear partial differential equations of the submarine pipeline vibration under the excitation of internal and external flow have the fourth-order partial derivatives at most in the spatial coordinates, the present invention combines the generalized finite difference method (GFDM) and the Newton-Raphson method (NR-GFDM), and applies them to solve the high-order nonlinear differential equations of the vortex-induced vibration of the submarine pipeline under the excitation of internal and external flow; and combines the Houbolt method to discretize the time partial derivatives in the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flow.

Description

Technical Field

[0001] The present invention belongs to the technical fields of submarine pipelines, modeling and analysis, and particularly relates to a numerical calculation method for studying a nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows. Background Art

[0002] With the rapid development of marine engineering technology, submarine water pipelines are gradually extending into deep waters, resulting in an increase in the pipeline's aspect ratio and enhanced nonlinear response. Although such slender pipelines are more conducive to transportation, the highly coupled torsional, axial, and lateral flexibility can easily exacerbate the vortex-induced vibration of the pipeline's free-span section. The main excitation sources for pipeline vortex-induced vibration are the fluid inside the pipe and the ocean current outside the pipe. Internal and external flows of different flow rates will produce different excitation forces, and the combined excitation effect of the two will make the dynamic characteristics and motion laws of submarine water pipelines more complex. Therefore, in order to effectively avoid catastrophic damage to pipelines in engineering practice, the mechanism of the internal and external fluid excitation effects and the pipeline vibration laws should be clarified. Therefore, in-depth research on the competitive or synergistic effects of internal and external fluid excitation on submarine water pipeline vibration is of great engineering significance.

[0003] The system of equations for vortex-induced vibration of submarine pipelines under internal and external flow excitation contains multiple nonlinear terms and fourth-order spatial and second-order temporal partial derivatives. The present invention proposes a numerical model that combines the generalized finite difference method with the Newton-Raphson method (NR-GFDM) iteration, and combines it with the Houbolt method for discretization to comprehensively solve the coupled variables in the system of equations. The generalized finite difference method (GFDM) is a new meshless method developed on the basis of the finite difference method. It retains the advantage of the finite difference method in converting partial differential equations into linear equations, and introduces the concept of point clusters. It is very advantageous when simulating complex boundary shapes and is a stable, efficient and accurate numerical simulation method. Summary of the Invention

[0004] Given the significant advantages of GFDM in solving partial differential equations, the present invention uses GFDM to introduce and solve a set of high-order nonlinear differential equations for vortex-induced vibration of submarine pipelines under internal and external flow excitation. Because GFDM is only applicable to nonlinear partial differential equations with a maximum order of second order, and the high-order nonlinear partial differential equations for submarine pipeline vibration under internal and external flow excitation have up to fourth-order partial derivatives in spatial coordinates, the present invention combines the generalized finite difference method (GFDM) with the Newton-Raphson method (NR-GFDM) to solve the set of high-order nonlinear differential equations for vortex-induced vibration of submarine pipelines under internal and external flow excitation; and combines the Houbolt method to discretize the time partial derivatives in the set of equations for vortex-induced vibration of submarine pipelines under internal and external flow excitation.

[0005] Based on the above concept, the purpose of the present invention is to provide a numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows, aiming to solve the coupling variables in the system of equations containing multiple nonlinear terms and high-order time-space partial derivatives, and to solve the nonlinear high-order partial differential equations in the vortex-induced vibration model of the submarine pipeline under the excitation of internal and external flows that vary with time and space.

[0006] To address the problem that the traditional generalized finite difference method cannot solve nonlinear terms, the present invention introduces the Newton-Raphson method into the generalized finite difference method (NR-GFDM). After improvement, it can iteratively solve the coupled nonlinear terms in the equation group. The method includes the following steps: non-dimensionalizing the vortex-induced vibration equation group of the submarine pipeline under internal and external flow excitation; discretizing the spatial partial differential terms in the pipeline vibration equation group using the generalized finite difference method; discretizing the time partial derivative terms in the pipeline vibration equation group using the Houbolt method; solving the discretized time partial derivative terms using the Euler method; substituting the obtained formula into the pipeline vibration equation group to establish a numerical model of the submarine water pipeline; continuing to discretize the boundary conditions using the generalized finite difference method; introducing the Newton-Raphson method to iteratively solve the unknown parameters of the equation group containing multiple nonlinear terms and high-order time-space partial derivatives at each moment. The present invention can solve the coupled variables in the equation group containing nonlinear high-order partial derivatives, and can be used to solve the nonlinear high-order partial differential equations that vary with time and space in the vortex-induced vibration model of the submarine pipeline under internal and external flow excitation.

[0007] The technical solution specifically adopted by the present invention to solve the technical problem is:

[0008] A numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows is proposed. The generalized finite difference method and the Newton-Raphson method are combined to solve the high-order nonlinear differential equations of the vortex-induced vibration of the submarine pipeline under the excitation of internal and external flows. The Houbolt method is then used to discretize the time partial derivatives in the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows.

[0009] Further, the following steps are included:

[0010] S1: Dimensionless the equations of vortex-induced vibration of submarine pipelines under internal and external flow excitation;

[0011] S2: The generalized finite difference method is used to discretize the spatial partial differential terms in the vortex-induced vibration equations of submarine pipelines under internal and external flow excitation;

[0012] S3: The Houbolt method is used to discretize the first-order and second-order partial derivatives in the time coordinate system of the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation;

[0013] S4: For the formulas obtained in S3 without initial conditions, use the Euler method to solve their initial conditions;

[0014] S5: Substitute the equations obtained in S2, S3 and S4 into the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation to establish a numerical model of the submarine water pipeline;

[0015] S6: Continue to discretize the boundary conditions using the generalized finite difference method;

[0016] S7: Introduce the Newton-Raphson method to iteratively solve the unknown parameters of the submarine pipeline vortex-induced vibration equation system at each moment under the excitation of internal and external flows.

[0017] Compared with the existing technology, the present invention and its preferred embodiment combine the generalized finite difference method and Newton-Raphson (NR-GFDM), which can solve the coupling variables in the system of equations containing nonlinear high-order partial derivatives, and can be applied to solve the nonlinear high-order partial differential equations that vary with time and space in the vortex-induced vibration model of submarine pipelines under internal and external flow excitation. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0019] Figure 1 This is a schematic diagram of a method flow in accordance with an embodiment of the present invention;

[0020] Figure 2 This is a spatial point distribution diagram of the generalized finite difference method according to an embodiment of the present invention;

[0021] Figure 3 This is a diagram showing the layout of different time layers according to an embodiment of the present invention;

[0022] Figure 4 is the bifurcation diagram of the pipeline lateral displacement under different ocean current velocities according to the embodiment of the present invention (U i =0);

[0023] Figure 5 is the bifurcation diagram of the pipeline axial displacement under different ocean current velocities according to the embodiment of the present invention (U i =0);

[0024] Figure 6 is the bifurcation diagram of the lateral vibration displacement of the pipeline midpoint under different ocean current velocities according to the embodiment of the present invention (U i =0,1,2,3.1). DETAILED DESCRIPTION

[0025] To make the features and advantages of this patent more clearly understood, the following embodiments are specifically described in detail as follows:

[0026] It should be noted that the following detailed description is illustrative and is intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meaning as commonly understood by those skilled in the art to which this application belongs.

[0027] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0028] like Figures 1-6 As shown, the calculation method of the nonlinear coupling system of the submarine flow pipeline provided by the embodiment of the present invention proposes a numerical model that combines the generalized finite difference method (GFDM) with the Newton-Raphson method (NR-GFDM) iteration, and combines it with the Houbolt method discretization to solve the coupling variables in the equation group as a whole.

[0029] The specific steps are as follows:

[0030] S1: Dimensionless the equations of vortex-induced vibration of submarine pipelines under internal and external flow excitation;

[0031] S2: The generalized finite difference method is used to discretize the spatial partial differential terms in the vortex-induced vibration equations of submarine pipelines under internal and external flow excitation;

[0032] S3: The Houbolt method is used to discretize the first-order and second-order partial derivatives in the time coordinate system of the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation;

[0033] S4: For the formulas obtained in S3 without initial conditions, use the Euler method to solve their initial conditions;

[0034] S5: Substitute the equations obtained in S2, S3 and S4 into the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation to establish a numerical model of the submarine water pipeline;

[0035] S6: Continue to discretize the boundary conditions using the generalized finite difference method;

[0036] S7: Since the numerical model established in S5 contains multiple nonlinear coupling terms at unknown moments, the Newton-Raphson method is introduced to iteratively solve the unknown parameters of the submarine pipeline vortex-induced vibration equation system under the excitation of internal and external flows at each moment.

[0037] Furthermore, the nonlinear differential equations for submarine pipeline vibration under the excitation of internal and external flow by S1 are:

[0038] Pipe axial vibration equation:

[0039]

[0040] Pipe lateral vibration equation:

[0041]

[0042] Where mi is the mass of the internal flow fluid per unit length, and the pipe structure damping c is s =2ζ(m p +m i +m a )ω n ,ω n is the natural frequency of the pipeline, ζ is the structural damping coefficient; the additional mass per unit length of the pipeline m a =πC a ρ e D p 2 / 4,C a is the additional mass coefficient, the lateral displacement of the pipe is w(x, t), the axial displacement is u(x, t), and the wake vortex vibration variable is q(x, t); the outer diameter of the pipe is D p , inner diameter is D i , the internal flow velocity is U i , the ocean current velocity is U e , A i is the internal flow cross-sectional area, p i A i is the internal flow pressure, the initial axial tension of the pipeline is T0, and E is the elastic modulus of the pipeline; I p is the moment of inertia of the pipe section, and the bending stiffness is EI p , g is the acceleration due to gravity, is the horizontal inclination of the seabed, m=m i +m p ; Partial differential terms in the system of equations represents inertial force; Damping force; Coriolis force representing internal flow; EI p w″″ represents the elastic restoring force; m i U i 2 w″, centrifugal force of internal flow; p i A i w″ represents the internal flow pressure that is considered; C D is the drag coefficient; represents the resistance along the unsteady internal flow, C L0 is the vortex lift coefficient when the structure is stationary, and the additional damping of the ocean current c f =C D ρe D p U e / 2; lift force F generated by ocean current flowing through the pipe L =C L0 ρ e U e 2 D p q / 4,ρ e is the density of the ocean current fluid.

[0043] Van der Pol wake vortex oscillator equation:

[0044]

[0045] Where ε and Λ are empirical coefficients, which can be adjusted by model calibration and experimental results and are taken as 0.3 and 12; q is the dimensionless wake oscillator, Ω f is the vortex shedding angular frequency.

[0046] The axial momentum equation of the internal flow is:

[0047]

[0048] Where, f s is the Darcy-Weisbach friction coefficient.

[0049] The dimensionless forms and dimensionless parameters of the following physical quantities are introduced:

[0050]

[0051] Substituting formula (5) into formulas (1) to (4), we get the dimensionless equations for pipeline vibration: Pipeline axial vibration equation:

[0052]

[0053] Pipe lateral vibration equation:

[0054]

[0055] Van der Pol wake vortex oscillator equation:

[0056]

[0057] The axial momentum equation of the internal flow is:

[0058]

[0059] Furthermore, S2 first distributes N points in the entire calculation area, and then converts the partial differential term on each node into a linear combination of the physical quantity of the neighboring points in the local sub-area of ​​each node and the weight coefficient. For the i-th node in the calculation area, select the n points closest to the node. s Points form a subregion. When the subregion of the i-th node is formed, the value of any point in the subregion is expanded using the Taylor series. The fourth-order Taylor series is truncated and the residual function B(Φ) is defined as follows:

[0060]

[0061] Where, Φ i,0 for i th Unknown value of the node, Φ i,j n s In the sub-region j th The unknown value of the node, h ij =x i，0 -x i,j Indicates i th Node and j th The vector distance between nodes, W(h ij ) is j th The weight function of the node.

[0062] Taking the minimum value of each order partial derivative in the residual function B(Φ), a linear system can be obtained. Solving the linear system yields each order partial derivative D Φ

[0063]

[0064] The expressions containing the first-order, second-order, and fourth-order spatial differential terms in the system of equations are extracted as follows:

[0065]

[0066]

[0067]

[0068] Furthermore, S3 uses the Houbolt method to discretize the time term of the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation, setting the time step Δt = t n+1 -t n , divide the time direction into equal intervals, and solve the physical quantity Φ of the unknown time layer through three known time layers n+1 , for the unknown physical quantity at time t=n+1, two reverse difference formulas are used to approximate it:

[0069]

[0070]

[0071] Furthermore, from formulas (15) and (16), it can be seen that when solving the physical quantity at time n+1, the physical quantities of t=n, n-1, and n-2 need to be known. However, the general initial conditions do not give the values ​​of t=n-1 and n-2. Therefore, S4 uses the Euler method to solve the first two initial values ​​of the Houbolt method.

[0072]

[0073]

[0074] Furthermore, S5 substitutes Equations (12) to (18) into the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation to obtain the following pipeline axial vibration equation:

[0075]

[0076] Pipe lateral vibration equation:

[0077]

[0078] Van der Pol wake vortex oscillator equation:

[0079]

[0080] The axial momentum equation of the internal flow is:

[0081]

[0082] Furthermore, the discretized S7 factor equations (19) to (22) also contain multiple nonlinear coupling terms at unknown times (t = n + 1). For the parameters to be determined at each time point in the entire system of equations, the Newton-Rapson method is used to iteratively solve them.

[0083] Φ k+1 =Φ k -J -1 (Φ k )f k (Φ k ) (twenty three)

[0084] Where, Φ k+1 and Φ k are the vectors of the values ​​of the k+1th and kth steps respectively; the matrix f is a nonlinear algebraic equation system consisting of four control equations and boundary conditions; J is the Jacobian matrix corresponding to f, and its elements are expressed as k is the number of iteration steps (k=0, 1, 2, ...).

[0085] When the loop iteration result meets the accuracy requirements, the iteration value is output. The iteration process and convergence conditions are as follows:

[0086]

[0087] max|Φ k+1 -Φ k |≤10 -6 (25)

[0088] During the iteration process, the Jacobian matrix is ​​configured as a sparse matrix to quickly solve for each variable. This cycle completes the calculation process.

[0089] In the above method, in view of the characteristics of the system of equations containing multiple nonlinear terms and fourth-order spatial and second-order temporal partial derivatives, a generalized finite difference method and a numerical model combined with the Newton-Raphson method (NR-GFDM) are proposed, and the generalized finite difference method and Houbolt method are used for discretization in space and time respectively to iteratively solve multiple nonlinear coupling variables in the system of equations as a whole. During the iterative process, since GFDM is obtained by generalizing the Taylor series expansion and the moving least squares method, the numerical discretization process is only related to the neighboring nodes on the central node. The Jacobian matrix is ​​configured as a sparse matrix, so that each variable can be solved quickly. By setting dimensionless parameters and variables, the dimensionless form of the equation is obtained, which improves the versatility of the equation. The natural frequency of the pipeline under different boundary conditions is solved and compared with the results of analytical solutions and other numerical methods to verify the accuracy of the model and method proposed in this embodiment.

[0090] In a specific application example of the present invention, Figure 2-Figure 6 As shown in the figure, a 100m long APIX65 grade steel pipeline system is taken as an example. The specific parameters are shown in the table below. Its aspect ratio is 307.69, which can better reflect the modal characteristics of the vortex-induced vibration of submarine pipelines with large aspect ratios.

[0091] Numerical example parameters

[0092] parameter Numerical parameter Numerical Pipe length L / m 100 <![CDATA[Initial lift coefficient C L0 > 0.3 <![CDATA[Outer diameter D p / m]]> 0.325 <![CDATA[Drag coefficient C D > 1.2 <![CDATA[Inner diameter D i / m]]> 0.305 <![CDATA[Additional mass coefficient C a > 1 Young's modulus E / (Gpa) 210 Seabed inclination angle α 0° <![CDATA[Pipe density ρ p / (kg / m 3 )]]> 7850 Strouhal number 0.2 <![CDATA[Seawater density ρ e / (kg / m 3 )]]> 1025 Vortex-induced vibration coupling coefficient Λ 12 <![CDATA[Inward flow density ρ i / (kg / m 3 )]]> 1000 Vortex-induced vibration coupling coefficient ε 0.3 <![CDATA[Aspect ratio L / D p > 307.69 <![CDATA[Mass ratio β = m i / (m i +m p )]]> 0.4584 <![CDATA[Axial tension T0 / N]]> 0 Pipeline structure damping 0.005

[0093] According to the research on the vibration of submarine water pipeline under the condition of simple support at both ends, it is known that the vibration of submarine water pipeline under the condition of simple support at both ends is not considered. i A i =0), the natural frequency of the system decreases with the increase of internal flow velocity, and the critical flow velocities at which the pipeline diverges in the form of first-order, second-order, third-order, and fourth-order modes are π, 2π, 3π, and 4π, respectively.

[0094] Unlike the internal flow velocity, in actual engineering, when the internal flow pressure changes slightly or the pipeline is short, its effect on the pipeline's natural frequency can be ignored. However, when the pipeline's length-to-diameter ratio is large or the internal flow velocity is high, the effect of the internal flow pressure change on the pipeline's natural frequency needs to be considered. Therefore, it is necessary to analyze the combined effect of internal flow pressure and internal flow velocity on the pipeline's natural frequency for a submarine water pipeline model that considers friction coupling between the pipe wall and the internal flow and axial vibration. When the dimensionless internal flow pressure p i A i =8, the natural frequency ratio of each order is p i A i =0, the critical flow velocity U of each stage in the pipeline i,cr Also decreased.

[0095] As the internal flow pressure and velocity increase, the pipeline's natural frequency decreases significantly. When the first-order natural frequency Im(w1) = 0, the pipeline's first unstable region appears, typically manifested by a gradual increase in vibration amplitude or jumps. At this point, the pipeline will gradually fatigue and fail. As the internal flow pressure and velocity continue to increase, the pipeline enters the second unstable region.

[0096] For the analysis of submarine pipeline vibration response under internal and external flow excitation, it is crucial to study the changes in its critical internal flow velocity to avoid pipeline instability and damage. With the increase of internal flow pressure, the first-order critical velocity of the submarine water pipeline with simple supports at both ends continues to decrease. When the dimensionless internal flow pressure increases to 9.78, the first-order critical velocity U i,cr = 0. The results show that when the internal flow pressure is considered, the internal flow pressure p i A i >0, the first-order critical flow velocity U of the pipeline i,cr is always less than π.

[0097] Those skilled in the art will appreciate that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to magnetic disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.

[0098] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the steps in the process. Figure 1 a process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0099] These computer program instructions may also be stored in a computer readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 a process or multiple processes and / or boxes Figure 1 The function specified in one or more boxes.

[0100] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing the instructions executed on the computer or other programmable device for implementing the process. Figure 1 a process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0101] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other manner. Any person skilled in the art may utilize the above-disclosed technical content to modify or modify the present invention into equivalent embodiments. However, any simple modifications, equivalent variations, and modifications to the above embodiments that do not depart from the technical content of the present invention and are based on the technical essence of the present invention remain within the scope of protection of the present invention.

[0102] This patent is not limited to the above-mentioned optimal implementation method. Anyone can derive various other forms of numerical calculation methods for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows under the inspiration of this patent. All equal changes and modifications made within the scope of the patent application of this invention should be covered by this patent.

Claims

1. A numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows, characterized in that: The generalized finite difference method and the Newton-Raphson method are combined to solve the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows; and the time partial derivatives of the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows are discretized by combining the Houbolt method. The following steps are involved: S1: Dimensionless the equations of vortex-induced vibration of submarine pipelines under internal and external flow excitation; S2: The generalized finite difference method is used to discretize the spatial partial differential terms in the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows; S3: The Houbolt method is used to discretize the first-order and second-order partial derivatives in the time coordinate system in the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows; S4: For the formulas obtained in S3 without initial conditions, the Euler method is used to solve their initial conditions; S5: Substitute the equations obtained from S2, S3 and S4 into the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows to establish a numerical model of the submarine water pipeline; S6: Continue to discretize boundary conditions using the generalized finite difference method; S7: The Newton-Raphson method is introduced to iteratively solve the unknown parameters of the submarine pipeline vortex-induced vibration equation system at each moment under the excitation of internal and external flows.

2. The numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows according to claim 1 is characterized by: In step S1, the nonlinear differential equations for submarine pipeline vibration under internal and external flow excitation are specifically: Pipe axial vibration equation: Pipe lateral vibration equation: Where mi is the mass of the internal flow fluid per unit length, and the pipeline structure damping c is s =2ζ(m p +m i +m a )ω n , ωn is the natural frequency of the pipeline, ζ is the structural damping coefficient; the additional mass per unit length of the pipeline m a =πC a ρ e D p 2 / 4,C a is the additional mass coefficient, the lateral displacement of the pipeline is w(x, t), the axial displacement is u(x, t), the wake vortex vibration variable is q(x, t); the outer diameter of the pipe is D p , inner diameter is D i , the internal flow velocity is U i , the ocean current velocity is U e , A i is the internal flow cross-sectional area, p i A i is the internal flow pressure, the initial axial tension of the pipeline is T0, and E is the elastic modulus of the pipeline; I p is the moment of inertia of the pipe section, and the bending stiffness is EI p , g is the acceleration due to gravity, is the horizontal inclination of the seabed, m=m i +m p ; Partial differential terms in the system of equations represents inertial force; Damping force; Coriolis force representing internal flow; EI p w″″ represents elastic recovery force; m i U i 2 w″, centrifugal force of internal flow; p i A i w″ represents the added internal flow pressure; C D is the drag force coefficient; represents the resistance along the unsteady internal flow, C L0 is the vortex lift coefficient when the structure is stationary, and the additional damping of the ocean current c f =C D ρ e D p U e / 2; lift F generated by ocean currents flowing through the pipe L =C L0 ρ e U e 2 D p q / 4,ρ e is the density of the ocean current fluid; L is the tube length; ρ i is the internal flow density; Van der Pol wake vortex oscillator equation: Where ε and Λ are empirical coefficients, which are determined and adjusted through model calibration and experimental results; q is the dimensionless wake oscillator, Ω f is the vortex shedding angular frequency; The internal flow axial momentum equation is: In the formula, f s is the Darcy-Weisbach friction coefficient; The dimensionless forms and dimensionless parameters of the following physical quantities are introduced: Substituting formula (5) into formulas (1) to (4), the dimensionless equations for pipeline vibration are obtained as follows: Pipe axial vibration equation: Pipe lateral vibration equation: Van der Pol wake vortex oscillator equation: The internal flow axial momentum equation is:

3. The numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows according to claim 2 is characterized by: Step S2 converts the dimensionless vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows into the following equations: formulas (6) to (9); GFDM is used to discretize the spatial partial differential terms; first, N points are arranged in the entire calculation area, and then the partial differential terms on each node are converted into a linear combination of the physical quantity of the neighboring points in the local sub-area of ​​each node and the product of the weight coefficient; for the i-th node in the calculation area, the n points closest to the node are selected. s points form a sub-region. When the sub-region of the i-th node is formed, the value of any point in the sub-region is expanded using the Taylor series. The fourth-order Taylor series is truncated and the residual function B(Φ) is defined as follows: In the formula, Φ i,0 for i th Unknown value of the node, Φ i,j n s In the sub-area j th The unknown value of the node, h ij =x i，0 -x i,j represents the vector distance between the ith node and the jth node, W(h ij ) is the weight function of the jth node; Taking the minimum value of each order partial derivative in the residual function B(Φ), we get a linear system with each order partial derivative D Φ By solving the linear system we get: The expressions containing the first-order, second-order and fourth-order spatial differential terms in the system of equations are extracted as follows:

4. The numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows according to claim 3 is characterized by: Step S3 uses the Houbolt method to discretize the time term of the vortex-induced vibration equations of the submarine pipeline under the excitation of internal and external flows, and sets the time step Δt=t n+1 -t n , the time direction is divided into equal intervals, and the physical quantity Φ of the unknown time layer is solved through three known time layers n+1 , for the unknown physical quantity at time t = n + 1, two reverse difference formulas are used, which can be approximately expressed as:

5. The numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows according to claim 4 is characterized by: In step S4, when considering solving the physical quantity at time n+1, it is necessary to know the physical quantities at t=n, n-1, and n-2, and use the Euler method to solve the first two initial values ​​of the Houbolt method; 6. The numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows according to claim 5 is characterized by: In step S5, equations (12) to (18) are substituted into the vortex-induced vibration equations of the submarine pipeline under internal and external flow excitation to obtain: Pipe axial vibration equation: Pipe lateral vibration equation: The axial momentum equation for internal flow is: Van der Pol wake vortex oscillator equation:

7. The numerical calculation method for studying the nonlinear vortex-induced vibration model of a submarine inclined pipeline under the coupling excitation of internal and external flows according to claim 6 is characterized by: In step S7, the numerically discretized equations (19) to (22) also contain nonlinear coupling terms at multiple unknown moments. The Newton-Raphson method is used to iteratively solve the unknown parameters of the entire system of equations at each moment: F k+1 =Φ k -J -1 (F k )f k (F k ) (23) In the formula, Φ k+1 and Φ k are the vectors of the values ​​of the k+1th and kth steps respectively; the matrix f is a nonlinear algebraic equation system consisting of four control equations and boundary conditions; J is the Jacobian matrix corresponding to f, and its elements are expressed as k is the number of iteration steps, k = 0, 1, 2, ...; When the loop iteration result meets the accuracy requirement, the iteration value is output. The iteration process and convergence conditions are as follows: max|Φ k+1 -F k |≤10 -6 (25) During the iteration process, the Jacobian matrix is ​​configured as a sparse matrix to quickly solve each variable; This cycle completes the calculation process.

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