A method for analyzing the tension characteristics of a gently sloping riser
By combining the Absolute Nodal Coordinates (ANCF) method and Euler–Bernoulli beam theory with the Morison equation, the accuracy and efficiency issues of tension characteristic analysis of gently sloping risers were solved, achieving high-precision riser tension analysis and optimized design of buoyancy sections.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- JIANGSU UNIV OF SCI & TECH
- Filing Date
- 2026-02-13
- Publication Date
- 2026-06-02
Smart Images
Figure CN122133387A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the technical field of dynamic analysis of marine engineering structures, and in particular to a method for analyzing the tension characteristics of a gently sloping riser. Background Technology
[0002] With the development of deep-sea oil and gas resources, risers, as key components connecting surface platforms and subsea wellheads, have attracted much attention regarding their dynamic response and fatigue performance. Ruffled-wave risers, by incorporating a buoyancy section in the middle of the pipe to form an arched configuration, can effectively reduce top tension and improve dynamic performance.
[0003] Existing riser analysis methods mainly include lumped mass method and finite element method, but they have limitations in terms of accuracy or computational efficiency when dealing with large deformations, geometric nonlinearities, and complex fluid structure interactions. The absolute nodal coordinate method (ANCF) describes element motion with global coordinates and is suitable for handling large displacement and rotation problems. It has been gradually applied to the analysis of marine cables and risers; however, its systematic analysis method for the tension characteristics of gently sloping risers is still imperfect. Summary of the Invention
[0004] Purpose of the invention: The purpose of this invention is to provide a method for analyzing the tension characteristics of a gently sloping riser, which can accurately predict the tension distribution of the riser in a complex marine environment.
[0005] Technical solution: To achieve the above objectives, the present invention provides a method for analyzing the tension characteristics of a wave-shaped riser, comprising the following steps: (1) Determine the structural parameters of the wave-shaped riser system and the marine environmental parameters; (2) In the coordinate system of the wave-shaped riser system, a three-dimensional model of the wave-shaped riser system is constructed. The riser is discretized into multiple reduced beam elements by using the absolute node coordinate method. The riser configuration is described by the node coordinates and gradient vectors in the global coordinate system. The node coordinate vectors of the reduced beam elements are obtained. The displacement field of any point in the reduced beam element is obtained by cubic Hermite interpolation. The mass matrix of the reduced beam element is derived. (3) Based on the Euler–Bernoulli beam theory, calculate the element strain energy corresponding to axial strain and bending strain respectively, and obtain the element generalized elastic force by differentiating the nodal coordinates. The element generalized elastic force includes the elastic force generated by the axial deformation of the element and the elastic force generated by the bending deformation of the element. (4) Calculate the distributed load and concentrated force acting on the riser. The distributed load and concentrated force include gravity, buoyancy, hydrodynamic load based on the Morison equation, and linear spring support force of the seabed contact section. Based on the principle of virtual work, the distributed load and concentrated force are converted into the generalized external force of the system. Based on the generalized external force and the generalized elastic force of the system, the generalized external force vector of the system is obtained. (5) Assemble the mass matrix, generalized external force vector and nodal coordinate vector of each unit according to the finite element rules to form the overall system dynamic equation, and express it in the form of a differential-algebraic equation system; (6) The system dynamic equations are solved by using the augmented method combined with numerical integration to obtain the absolute node coordinate vectors of the riser at different times, and then the tension along the entire length of the riser is calculated.
[0006] Optionally, in step (1), the wave-shaped riser system includes a wave-shaped riser, a platform, and a seabed. The upper end of the wave-shaped riser is connected to the platform, and the lower end of the wave-shaped riser is connected to the seabed. The wave-shaped riser includes a suspended section, a buoyancy section, a descent section, and a ground-lying section. The contact point between the wave-shaped riser and the seabed is the contact point, and the riser between the contact point and the lower end is the ground-lying section. The structural parameters of the wave-shaped riser include the coordinates of the upper end and the lower end, the riser length, the riser outer diameter, the riser wall thickness, the material density, the bending stiffness, the starting position of the buoyancy section, the arrangement length of the buoyancy section, the drag coefficient, and the inertia coefficient. The marine environmental parameters include water depth, seawater density, ocean current velocity, and wave parameters. The wave parameters include wave height, period, and incident direction.
[0007] Optionally, in step (1), the starting position and arrangement length of the buoyancy section are set according to the buoyancy ratio. The buoyancy ratio is defined as the ratio of the net buoyancy of the buoyancy section to the net underwater weight of the riser, and the value range is 10%–50%. The equivalent buoyancy distribution of the buoyancy section is calculated based on the buoyancy ratio, and the arrangement range of the buoyancy section in the riser axis is determined.
[0008] Optionally, the buoyancy section has a length of 150~250m and its starting position is located in the range of 700~900m from the upper end point.
[0009] Optionally, the coordinate system of the wave riser system in step (2) is as follows: the center point where the platform intersects with the sea level is the origin of the XYZ coordinate system, the sea level is the XY plane, the ocean current direction is set along the positive X-axis, and the Z-axis is perpendicular to the XY plane; the top surge response of the platform under wave action is simulated by applying simple harmonic motion along the X-axis direction at the top of the platform.
[0010] Optionally, step (2) specifically includes the following steps: The nodal coordinate vector of a reduced beam element consists of the global position vectors of the nodes at both ends of the element and their first-order gradient vectors with respect to the local coordinates of the element. The global position vector of any point on the element is obtained by interpolating the nodal coordinate vectors using a cubic Hermite shape function. The nodal coordinates of the reduced beam element consist of the positions of its two end nodes and their derivatives with respect to the local coordinates of the element. Under the assumption of a rigid section, the length is... The nodal coordinate vector of the reduced beam element for: , in, To reduce the beam element The coordinate vector at the starting point of the pipeline, To reduce the beam element The coordinate vector at the end of the pipeline. for The position and direction vector at the starting point of the pipeline. for The position and direction vector at the end of the pipeline. for Tangent direction vector at the starting point of the pipe. for The tangent direction vector at the end of the pipe. for The absolute position coordinates of the pipeline starting point in the x-direction. for The absolute position coordinates of the pipeline starting point in the y-direction. for The absolute position coordinates of the pipeline starting point in the z-direction. for The absolute position coordinates of the pipeline starting point in the x-direction. for The absolute position coordinates of the pipeline starting point in the y-direction. for The absolute position coordinates of the pipeline starting point in the z-direction. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate x of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate y of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the element's axial coordinate z. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate x of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate y of the element. for The partial derivative of the position vector at the starting point of the pipeline with respect to the axial coordinate z of the element; For the global position vector of any point P on the axis of the reduced beam element The following is obtained through cubic Hermite interpolation of the absolute node coordinates of the element: , in, To reduce the absolute nodal coordinate vector of the beam element, which is time-dependent; It is a three-point quadratic Hermitian function, expressed as follows: , Where I3 is the third-order identity matrix. For dimensionless parameters, To reduce the length of the beam element; Since the absolute node coordinates of the element are only in time... The function, therefore the corresponding global speed and acceleration vector It can be represented as: , To reduce the absolute nodal coordinate vector of beam elements The derivative of To reduce the absolute nodal coordinate vector of beam elements The second derivative; Similarly, the derivative of the position vector of a point on an element with respect to the local coordinates of the element can be expressed as: , in, Represents global position coordinates The first partial derivative with respect to the local coordinate x, Represents the shape function matrix The first partial derivative with respect to the local coordinate x, Represents global position coordinates The second-order partial derivative with respect to the local coordinate x, Represents the shape function matrix The second partial derivative with respect to the local coordinate x; The kinetic energy expression of the element is derived from the velocity vector of a point on the element: , in, Represented as the cross-sectional area of the element. For unit density, The unit mass matrix, The global velocity vector at a point on the element. Let be the transpose of the global velocity vector at a point on the element. Let be the velocity vector of a point on the element. Let be the transpose of the velocity vector at a point on the element. is the transpose matrix of the shape functions of the element; when When determined, The element mass matrix is a constant matrix, and its value is derived from the kinetic energy expression. : , in, , M 21 , The unit mass matrix M is divided into four 6×6 matrices.
[0011] Optionally, step (3) specifically includes the following steps: Axial strain is calculated using the norm of the first derivative of the element position vector with respect to the local coordinates; bending curvature is calculated using the cross product of the first and second derivatives of the element position vector with respect to the local coordinates; and generalized elastic force is obtained by taking the partial derivative of the strain energy with respect to the element nodal coordinate vector. Strain energy of riser unit The expression is: , Where E is the elastic modulus of the material, and J is the moment of inertia of the beam section; For axial strain, it is expressed as: , in, Let x be the first partial derivative of the position vector with respect to the local coordinate x. Let be the transpose of the first-order partial derivative of the position vector with respect to the local coordinate x; For curvature, it is expressed as: , Strain energy of riser unit For the coordinate vector of the unit node By taking the partial derivatives, we can obtain the generalized elastic force of the element generated by the strain energy. , is represented as: , in, The elastic force generated by the axial deformation of the unit. The elastic force generated by the bending deformation of the unit; , , in For curvature absolute node coordinate vector The partial derivatives yield the transpose matrix.
[0012] Optionally, step (4) specifically includes the following steps: Gravity load on riser unit Represented as: , The load of buoyancy acting on the riser unit is expressed as: , in The density of seawater, It is the acceleration due to gravity; Hydrodynamic loads calculated based on the Morison equation include tangential drag force, normal drag force, and inertial force. The nonlinear drag in the Morison equation is divided into tangential force and normal force. Let the tangential unit vector of the pipe element be: , For any vector a, its tangential projection onto the riser unit is: Then its normal projection is The normal projection transformation matrix is obtained. The expression is: , In the Morison equation, the tangential drag force and normal drag force They are represented as follows: , , in, The diameter of the riser pipe. The density of seawater, For relative velocity, The tangential component of the relative velocity, The normal component of the relative velocity. For tangential loading function, This is the normal load function; This is the drag force coefficient; Among them, the drag coefficient Based on semi-empirical formulas and Reynolds numbers calculated using normal relative velocity and riser diameter. Related determination, represented as: , Tangential loading function and normal loading function Represented as: , in, The relative velocity of the ocean current; Tangential component of relative velocity and the normal component of relative velocity Represented as: , Seabed forces are simulated using a linear spring model to represent the normal support force on the seabed. For the contact point, the linear spring support force... for: , in, The depth of intrusion; Equivalent stiffness; Seabed depth; The unit direction vector in the Z direction; Let P be the position vector of any point; Obtain the generalized force vector The generalized force vector includes the system's generalized elastic force. and system generalized external forces , , Among them, the generalized external force of the system The expression is: , in The load on the riser unit is caused by gravity. buoyancy load on riser unit Tangential drag force Normal drag force and linear spring support force constitute.
[0013] Optionally, step (5) specifically includes the following steps: The system of dynamic differential algebraic equations for the constrained system is established using the Lagrange multiplier method, and its form is as follows: The dynamic equations of the ANCF mooring cable are a set of index-3 differential-algebraic equations. Solving them using the Baumgarte method and rewriting the formulas in matrix form yields a set of index-1 differential-algebraic equations. , in, The system's quality matrix, For constraint equations For generalized coordinates Jacobi matrix, for The transpose of the matrix, Let be the system acceleration vector. For Lagrange multipliers, It is a generalized force vector. The right-hand side of the constraint acceleration equation includes velocity nonlinear terms and explicit time derivative terms; The system constraint equations The first and second partial derivatives with respect to time are as follows: , , in, To constrain the Jacobian matrix for generalized coordinates, To constrain the explicit partial derivatives with respect to time, For velocity vectors, For acceleration vectors, To constrain the second-order explicit partial derivatives with respect to time, for right The partial derivatives, These are mixed partial derivatives, i.e., constraint pairs. Again The partial derivative; The right-hand side of the constrained acceleration equation is obtained from the above equation. : , In the formula: , We obtain a set of ordinary differential equations, namely the system dynamics equations: .
[0014] Optionally, step (6) specifically includes the following steps: The augmented method and the introduction of the Baumgarte default stabilization term are used to numerically solve the dynamic differential-algebraic equation system, transforming the differential-algebraic equation system of index -3 into a system of first-order ordinary differential equations that can be directly integrated. To address potential breaches of the equations during numerical computation, a breach correction method based on control feedback is adopted, which involves introducing a feedback system into the constraint equations. , in, The right-hand side of the constraint equations after the Baumgarte method correction is represented. , and are stability coefficients. Empirical values for and are between 5 and 50. When = , the stable response is fast, resulting in a system of first-order differential equations: , In the formula, B represents the first N on the right side. dof Line, i.e., the system acceleration vector, N dof Let be the total number of degrees of freedom of the system; therefore, the system's velocity and position vectors can be obtained by solving the system's differential equations; By solving the matrix-form equations of motion dynamically, the absolute nodal coordinate vectors of the riser unit are obtained. ,pass The position vector of the riser unit during the simulation process can be calculated. Elastic force The elastic force generated by the axial deformation of the riser unit This refers to the tension of the riser.
[0015] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: The present invention is the first to systematically introduce the Absolute Nodal Coordinates (ANCF) method to reduce the beam element system into the analysis of the tension characteristics of the wave-shaped riser. It uses the nodal position vector and its first-order gradient vector with respect to the local coordinates of the element as the nodal coordinate vector, and realizes high-precision interpolation of the global position vector of any point in the element through the cubic Hermite shape function. This modeling method does not require the introduction of rotational coordinate linearization approximation, fundamentally avoiding the rotational singularity problem in the large rotation analysis of the traditional finite element method, and can accurately describe the large displacement, large rotation and finite tensile deformation of the riser. Attached Figure Description
[0016] Figure 1 A schematic diagram of the wave-shaped riser system provided by the present invention; Figure 2 A three-dimensional riser unit model diagram provided by the present invention; Figure 3 A schematic diagram of the initial shape of the catenary riser and the gently sloping riser provided by the present invention; Figure 4 A tension diagram of the catenary riser and the gently sloping riser provided by the present invention; Figure 5 A schematic diagram of the initial shape of the wave-shaped riser with different lengths of buoyancy sections provided by the present invention; Figure 6 A schematic diagram of the tension of a wave-shaped riser with different lengths of buoyancy sections provided by the present invention; Figure 7 A schematic diagram of the initial shape of the wave-shaped riser at different starting positions of the buoyancy section provided by the present invention; Figure 8 A schematic diagram of the tension of the wave-shaped riser at different starting positions of the buoyancy section provided by the present invention; Figure 9 A schematic diagram illustrating the change of tension over time at the upper endpoint of different wave incident angles provided by the present invention; Figure 10 A schematic diagram showing the change of tension at the point of contact with different wave incident angles over time, provided by the present invention. Figure 11 This is a schematic diagram showing the tension change over time at the lower end of different wave incident angles, as provided by the present invention. Detailed Implementation
[0017] The technical solution of the present invention will be further described below with reference to the accompanying drawings.
[0018] like Figure 1 As shown, the present invention provides a method for analyzing the tension characteristics of a gently sloping riser, comprising the following steps: (1) Determine the structural parameters and marine environmental parameters of the wave-shaped riser system. The wave-shaped riser system includes the wave-shaped riser, platform, and seabed. The upper end of the wave-shaped riser is connected to the platform, and the lower end of the wave-shaped riser is connected to the seabed. The wave-shaped riser includes a suspended section, a buoyancy section, a descent section, and a ground-lying section. The contact point between the wave-shaped riser and the seabed is the contact point, and the riser between the contact point and the lower end is the ground-lying section. The structural parameters of the wave-shaped riser include the coordinates of the upper end and the lower end, the riser length, the riser outer diameter, the riser wall thickness, the material density, the bending stiffness, the starting position of the buoyancy section, the arrangement length of the buoyancy section, the drag coefficient, and the inertia coefficient. The marine environmental parameters include water depth, seawater density, ocean current velocity, and wave parameters. The wave parameters include wave height, period, and incident direction. The specific parameter table is shown in Table 1. In addition, the polar moment of inertia of the cross section J = 5.4094 × 10 -4 m 4 Elastic modulus E = 4.9001 × 10 4 MPa.
[0019] Table 1 Parameter Table
[0020]
[0021] The coordinate system of the wave-shaped riser system is set up, with the center point where the platform intersects the sea level as the origin of the XYZ coordinate system, the sea level as the XY plane, the ocean current direction set along the positive X-axis, and the Z-axis perpendicular to the XY plane; the top swell response of the platform under wave action is simulated by applying a simple harmonic motion along the X-axis at the top of the platform.
[0022] The starting position and arrangement length of the buoyancy section are determined based on the buoyancy ratio. The buoyancy ratio is defined as the ratio of the net buoyancy of the buoyancy section to the net underwater weight of the riser, with a value range of 10%–50%. The equivalent buoyancy distribution of the buoyancy section is calculated based on the buoyancy ratio, and the arrangement range of the buoyancy section in the riser axis is determined.
[0023] In the design of the gently sloping riser, the length of the buoyancy section should be selected within the range of 150 to 250 meters, and the starting position is recommended to be within the range of 700 to 900 meters from the upper end.
[0024] Specifically, increasing the arrangement length significantly improves net buoyancy and increases the riser waveform curvature, which helps reduce top tension. However, it also increases local tension and bending curvature at the end of the buoyancy section and adjacent descending sections. Lowering the starting position of the buoyancy section expands the horizontal span of the riser, making the configuration more gradual, thus increasing tension at the upper end, while correspondingly reducing tension and maximum curvature at the end of the buoyancy section. Therefore, the arrangement length and starting position need to be optimized synergistically to balance the overall tension distribution, control local bending stress, and effectively block the transmission of upper dynamic response to the contact area, ultimately improving the safety and fatigue performance of the gentle-wave riser system under complex sea conditions.
[0025] (2) Construct a three-dimensional model of the wave-shaped riser system, and use the absolute node coordinate method to discretize the riser into multiple reduced beam elements, where the length of a single reduced beam element is 20~30m; and describe the riser configuration with node coordinates and gradient vectors in the global coordinate system, obtain the displacement field of any point in the reduced beam element through cubic Hermite interpolation, and derive the mass matrix of the reduced beam element. The nodal coordinate vector of the reduced beam element consists of the global position vector of the nodes at both ends of the element and its first-order gradient vector with respect to the local coordinates of the element. The global position vector of any point on the element is obtained by interpolating the nodal coordinate vector using a cubic Hermite shape function. The nodal coordinates of a reduced beam element consist of the positions of its two end nodes and their derivatives with respect to the element's local coordinates. Under the assumption of a rigid section, the length is... The nodal coordinate vector of the reduced beam element for:
[0026] in, To reduce the beam element The coordinate vector at the starting point of the pipeline, To reduce the beam element The coordinate vector at the end of the pipeline. for The position and direction vector at the starting point of the pipeline. for The position and direction vector at the end of the pipeline. for Tangent direction vector at the starting point of the pipe. for The tangent direction vector at the end of the pipe. for The absolute position coordinates of the pipeline starting point in the x-direction. for The absolute position coordinates of the pipeline starting point in the y-direction. for The absolute position coordinates of the pipeline starting point in the z-direction. for The absolute position coordinates of the pipeline starting point in the x-direction. for The absolute position coordinates of the pipeline starting point in the y-direction. for The absolute position coordinates of the pipeline starting point in the z-direction. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate x of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate y of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the element's axial coordinate z. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate x of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate y of the element. for The partial derivative of the position vector at the starting point of the pipeline with respect to the axial coordinate z of the element; Give the global position vector of any point P on the central axis of its initial configuration and its deformed configuration, such as Figure 2 The diagram illustrates the initial and deformed forms of the riser unit; before and after deformation, the global position vector of any point P on the axis of the reduced beam unit is shown. The following is obtained through cubic Hermite interpolation of the absolute node coordinates of the element:
[0027] in, To reduce the absolute nodal coordinate vector of the beam element, which is time-dependent; It is a three-point quadratic Hermitian function, expressed as follows:
[0028] Where I3 is the third-order identity matrix. For dimensionless parameters, To reduce the length of the beam element; Since the absolute node coordinates of the element are only in time... The function, therefore the corresponding global speed and acceleration vector It can be represented as: , To reduce the absolute nodal coordinate vector of beam elements The derivative of To reduce the absolute nodal coordinate vector of beam elements The second derivative; Similarly, the derivative of the position vector of a point on an element with respect to the local coordinates of the element can be expressed as: , in, Represents global position coordinates The first partial derivative with respect to the local coordinate x, Represents the shape function matrix The first partial derivative with respect to the local coordinate x, Represents global position coordinates The second-order partial derivative with respect to the local coordinate x, Represents the shape function matrix The second partial derivative with respect to the local coordinate x; The kinetic energy expression of the element is derived from the velocity vector of a point on the element:
[0029] in, Represented as the cross-sectional area of the element. For unit density, The unit mass matrix, The global velocity vector at a point on the element. Let be the transpose of the global velocity vector at a point on the element. Let be the velocity vector of a point on the element. Let be the transpose of the velocity vector at a point on the element. is the transpose matrix of the shape functions of the element; when When determined, The element mass matrix is a constant matrix, and its value is derived from the kinetic energy expression. :
[0030] in, , M 21 , The unit mass matrix M is divided into four 6×6 matrices. (3) Based on the Euler–Bernoulli beam theory, calculate the element strain energy corresponding to axial strain and bending strain respectively, and obtain the element elastic force by differentiating the nodal coordinates;
[0031] In step (3), the axial strain is calculated by the norm of the first derivative of the element position vector with respect to the local coordinates, the bending curvature is calculated by the cross product of the first and second derivatives of the element position vector with respect to the local coordinates, and the generalized elastic force is obtained by taking the partial derivative of the strain energy with respect to the element node coordinate vector. Strain energy of riser unit The expression derivation is as follows:
[0032] Where E is the elastic modulus of the material, and J is the moment of inertia of the beam section; For axial strain, it is expressed as:
[0033] in, Let x be the first partial derivative of the position vector with respect to the local coordinate x. Let be the transpose of the first-order partial derivative of the position vector with respect to the local coordinate x; For curvature, it is expressed as:
[0034] Strain energy of riser unit For the coordinate vector of the unit node By taking the partial derivatives, we can obtain the generalized elastic force of the element generated by the strain energy. , is represented as:
[0035] in, The elastic force generated by the axial deformation of the unit. The elastic force generated by the bending deformation of the unit;
[0036]
[0037] in For curvature absolute node coordinate vector The partial derivatives yield the transpose matrix;
[0038] (4) Calculate the distributed load and concentrated force acting on the riser. The distributed load and concentrated force include gravity, buoyancy, hydrodynamic load based on the Morison equation, and linear spring support force of the seabed contact section. Based on the principle of virtual work, the distributed load and concentrated force are converted into the generalized external force of the system. Based on the generalized external force and the generalized elastic force of the system, the generalized force vector of the system is obtained. Step (4) specifically includes the following steps: Gravity load on riser unit It can be represented as:
[0039] The load of buoyancy acting on the riser unit is expressed as:
[0040] in The density of seawater, It is the acceleration due to gravity; Hydrodynamic loads calculated based on the Morison equation include tangential drag force, normal drag force, and inertial force. The nonlinear drag in the Morison equation is divided into tangential force and normal force. Let the tangential unit vector of the pipe element be:
[0041] For any vector a, its tangential projection onto the riser unit is: Then its normal projection is The normal projection transformation matrix is obtained. :
[0042] In the Morison equation, the tangential drag force and normal drag force They are represented as follows:
[0043]
[0044] in, The diameter of the riser pipe. The density of seawater, For relative velocity, The tangential component of the relative velocity, The normal component of the relative velocity. For tangential loading function, This is the normal load function; This is the drag force coefficient; Among them, the drag coefficient Based on semi-empirical formulas and Reynolds numbers calculated using normal relative velocity and riser diameter. Related determination, meaning:
[0045] Tangential loading function and normal loading function Represented as:
[0046] in, The relative velocity of the ocean current; Tangential component of relative velocity and the normal component of relative velocity Represented as:
[0047] Seabed forces are simulated using a linear spring model to represent the normal support force on the seabed. For the contact point, the linear spring support force f... z for:
[0048] Where Δz is the intrusion depth; k is the equivalent stiffness; Z bed seabed depth; e z is the unit direction vector in the Z direction; rp is the position vector of any point P.
[0049] Obtain the generalized force vector The generalized force vector includes the system's generalized elastic force. and system generalized external forces ,
[0050] Among them, the generalized external force of the system The expression is:
[0051] in It is by , , , and constitute; (5) Combine the mass matrix M and the generalized external force vector of each unit. and node coordinate vector Assemble according to the finite element rules to form the overall system dynamic equations, and express them in the form of a differential-algebraic equation system. In step (5), the dynamic differential algebraic equations of the constrained system are established using the Lagrange multiplier method, and their form is as follows: The dynamic equations of the ANCF mooring cable are a set of index-3 differential-algebraic equations. Solving them using the Baumgarte method and rewriting the formulas in matrix form yields a set of index-1 differential-algebraic equations.
[0052] in, The system's quality matrix, For constraint equations For generalized coordinates Jacobi matrix, for The transpose of the matrix, Let be the system acceleration vector. For Lagrange multipliers, It is a generalized force vector. The right-hand side of the constraint acceleration equation includes velocity nonlinear terms and explicit time derivative terms; The system constraint equations The first and second partial derivatives with respect to time are as follows:
[0053]
[0054] in, To constrain the Jacobian matrix for generalized coordinates, To constrain the explicit partial derivatives with respect to time, For velocity vectors, For acceleration vectors, To constrain the second-order explicit partial derivatives with respect to time, for right The partial derivatives, These are mixed partial derivatives, i.e., constraint pairs. Again The partial derivative; The right-hand side of the constrained acceleration equation is obtained from the above equation. :
[0055] In the formula:
[0056] We obtain a set of ordinary differential equations, namely the system dynamics equations: .
[0057] (6) The augmented method combined with numerical integration is used to solve the system dynamic equations to obtain the absolute node coordinate vectors of the riser at different times, and then the tension along the entire length of the riser is calculated. In step (6), the augmentation method is used and the Baumgarte default stabilization term is introduced to numerically solve the dynamic differential algebraic equation system, transforming the differential algebraic equation system of index-3 into a first-order ordinary differential equation system that can be directly integrated. To address potential breaches of the equations during numerical computation, a breach correction method based on control feedback is adopted, which involves introducing a feedback system into the constraint equations.
[0058] in, The right-hand side of the constraint equations after the Baumgarte method correction is represented. , and are stability coefficients. Empirical values for and are between 5 and 50. When = , the stable response is fast, resulting in a system of first-order differential equations:
[0059] In the formula, B represents the first N on the right side. dof Line, i.e., the system acceleration vector, N dof Let be the total number of degrees of freedom of the system; therefore, the system's velocity and position vectors can be obtained by solving the system's differential equations; The ode113 solver was used to solve the matrix-form equations of motion dynamically, yielding the absolute nodal coordinate vectors of the riser unit. ,pass The position vector of the riser unit during the simulation process can be calculated. Elastic force The elastic force generated by the axial deformation of the riser unit This refers to the tension of the riser.
[0060] like Figure 3 The diagram shows the initial shape of the catenary riser and the wave-shaped riser provided by the present invention. The main difference between the wave-shaped riser and the catenary riser in terms of shape lies in the presence of the buoyancy section. Due to the additional buoyancy, the wave-shaped riser forms a clear transition between the descending section and the upper section, while the catenary riser presents a smoother continuous curve shape.
[0061] like Figure 4 The diagram illustrates the tension of the catenary riser and the wave-shaped riser provided by this invention. In the catenary layout, the tension decreases continuously along the riser's length. However, in the wave-shaped layout, due to the buoyancy section, the tension first decreases, then increases, and then decreases again, resulting in a maximum tension at the end of the buoyancy section. A comparison reveals that the maximum tension of the riser in both layouts occurs at the top end, with the maximum tension in the wave-shaped layout being significantly smaller than that in the catenary layout.
[0062] like Figure 5 The diagram shows the initial shape of the wave-shaped riser with different lengths of buoyancy section provided by the present invention. The longer the buoyancy section, the greater the buoyancy it provides, and the steeper the wave of the riser.
[0063] like Figure 6The diagram illustrates the tension of a gently sloping riser with different buoyancy sections provided by this invention. As the buoyancy section increases, the tension at the top end decreases, while the tension at the end of the buoyancy section and on the descending section increases. This is because the increase in buoyancy reduces the tension requirement at the top end of the riser; however, the increase in buoyancy causes the descending section of the riser to lengthen, thus increasing the tension at the end of the buoyancy section and on the descending section.
[0064] like Figure 7 The diagram shows the initial shape of the wave-shaped riser at different starting positions of the buoyancy section provided by the present invention. The farther the starting position of the buoyancy section is from the upper end point, the greater the span between the upper end point of the riser and the bottom contact point, and the gentler the wave of the riser.
[0065] like Figure 8 The diagram shows the tension of the wave-shaped riser at different starting positions of the buoyancy section provided by the present invention. As the starting position of the buoyancy section is farther from the upper end point, the tension at the upper end point increases. At the same time, as the starting position of the buoyancy section becomes farther away, the length of the descending section becomes shorter, which in turn leads to a decrease in the tension at the end of the buoyancy section.
[0066] like Figure 9 The diagram shows the tension at the upper end point of the riser at different wave incident angles as provided by the present invention. As the wave propagation direction changes, the dynamic tension of the riser exhibits strong nonlinearity. When the wave incident direction changes from 0° to 90° to 180°, the tension at the upper end point of the riser first decreases by about 8 kN and then increases by about 6 kN.
[0067] like Figure 10 The diagram shows the tension at the contact point under different wave incidence angles as provided by this invention over time. As the wave propagation direction changes, the tension at the contact point first decreases by approximately 0.19 kN and then increases by approximately 0.06 kN. When the wave incidence angle changes while other conditions remain constant, the angle at which the maximum tension occurs at the contact point coincides with the riser arrangement direction, but the change in tension is not as significant as at the upper end point.
[0068] like Figure 11 The diagram shows the tension at the lower end of the riser at different wave incident angles over time, as provided by this invention. As the wave propagation direction changes, the tension at the lower end of the riser first decreases by about 0.15 kN and then increases by about 0.06 N. When the wave incident angle changes while other conditions remain constant, the angle at which the maximum tension occurs coincides with the riser arrangement direction, but the change in tension is not as significant as at the upper end.
[0069] The core difference between the ANCF reduced beam element model constructed in this invention and the lumped mass method used in commercial software lies in the following: The absolute nodal coordinate method uses the nodal position vector and its first-order gradient vector with respect to the element's local coordinates as the nodal coordinate vector, accurately describing the global position field of any point within the element through consecutive cubic Hermite shape functions. In contrast, the lumped mass method discretizes the riser into mass points and connects them through spring-damped elements, making it difficult to accurately characterize the continuous bending configuration and local curvature changes of the structure at low element density. Therefore, this invention requires only 90 ANCF reduced beam elements to obtain convergent tension analysis results; in comparison, the lumped mass method typically requires approximately 360 elements for analysis of equivalent accuracy. This invention achieves the same or even better computational accuracy with about one-quarter the number of elements as the lumped mass method, significantly improving computational efficiency and fully demonstrating the high efficiency advantage of this method while ensuring accuracy.
[0070] In terms of environmental load and boundary condition modeling, this invention fully integrates a hydrodynamic load model based on the Morison equation within the ANCF framework. Through tangential unit vectors and normal projection transformation matrices, the nonlinear drag force is accurately decomposed into tangential and normal drag force components. A semi-empirical formula related to the Reynolds number is introduced to determine the drag force coefficient, making the hydrodynamic calculations more closely reflect actual sea conditions. A linear spring model is used for the seabed contact section. The normal support force is directly calculated based on the intrusion depth, equivalent stiffness, and seabed depth, and updated synchronously with the riser configuration, achieving continuous and stable simulation of the dynamic response in the contact area. This integrated strategy effectively overcomes the technical limitations of traditional methods, such as linearizing and equivalencing fluid loads and performing offline post-processing of seabed contact.
[0071] For the index-3 differential-algebraic equations described by the ANCF dynamic equations, this invention employs an augmented method and introduces a Baumgarte violation stabilization term for numerical solution. By introducing feedback corrections (stability coefficients α and β) for position-level and velocity-level constraint violations into the right-hand side of the constraint acceleration equations, the original index-3 DAEs are transformed into a directly integrable set of first-order ordinary differential equations, effectively suppressing constraint drift during long-term numerical integration. Compared with directly integrating the unstabilized equations, this invention significantly improves computational stability and simulation duration adaptability at the same time step, making it particularly suitable for long-term dynamic response analysis under wave-platform coupled excitation.
[0072] Based on the aforementioned high-precision simulation platform, this invention, for the first time, systematically investigated the influence of multiple factors, including buoyancy section length, starting position, wave incidence angle, wave height, and wave period, on the effective tension distribution and fatigue damage of gently sloping risers. Quantitatively, it revealed that increasing the buoyancy section length reduces the effective tension at the suspension point but increases the local tension peak at the end of the buoyancy section and the descending section; shifting the starting position of the buoyancy section further back makes the riser configuration more gradual, but the effective tension at the suspension point increases accordingly; the wave incidence angle has a significant nonlinear effect on the dynamic tension amplitude at the suspension point, contact point, and anchoring point, with the angle of maximum tension coinciding with the riser's layout direction. These findings provide direct and reliable data support for the optimized layout of the buoyancy section of gently sloping risers and the adaptive assessment of the wave incidence direction, filling a gap in existing research on the collaborative design of buoyancy section and environmental parameters.
[0073] Furthermore, the method of this invention does not rely on specific commercial software; the entire solution process can be independently implemented on general numerical platforms such as MATLAB, facilitating customized extensions by engineering technicians according to actual project needs. The extracted nodal displacements, velocities, and accelerations can be directly used for subsequent fatigue damage accumulation calculations based on the Miner criterion, forming an integrated analysis chain from environmental load input and dynamic response solution to fatigue life assessment. This provides an autonomous, controllable, efficient, and reliable technical tool for the safe design, optimized layout, and integrity management of deep-sea riser systems.
[0074] In summary, this invention significantly outperforms existing technologies in terms of model accuracy, adaptability to geometric nonlinearity, completeness of fluid-structure interaction and seabed contact modeling, numerical solution stability, depth of parameter analysis, and engineering practicality. It has important theoretical value and broad engineering application prospects for promoting the transformation of the dynamic design method for gently sloping risers from empirical analogy to refined numerical simulation.
Claims
1. A method for analyzing the tension characteristics of a gently sloping riser, characterized in that, Includes the following steps: (1) Determine the structural parameters of the wave-shaped riser system and the marine environmental parameters; (2) In the coordinate system of the wave-shaped riser system, a three-dimensional model of the wave-shaped riser system is constructed. The riser is discretized into multiple reduced beam elements by using the absolute node coordinate method. The riser configuration is described by the node coordinates and gradient vectors in the global coordinate system. The node coordinate vectors of the reduced beam elements are obtained. The displacement field of any point in the reduced beam element is obtained by cubic Hermite interpolation. The mass matrix of the reduced beam element is derived. (3) Based on the Euler–Bernoulli beam theory, calculate the element strain energy corresponding to axial strain and bending strain respectively, and obtain the element generalized elastic force by differentiating the nodal coordinates. The element generalized elastic force includes the elastic force generated by the axial deformation of the element and the elastic force generated by the bending deformation of the element. (4) Calculate the distributed load and concentrated force acting on the riser. The distributed load and concentrated force include gravity, buoyancy, hydrodynamic load based on the Morison equation, and linear spring support force of the seabed contact section. Based on the principle of virtual work, the distributed load and concentrated force are converted into the generalized external force of the system. Based on the generalized external force and the generalized elastic force of the system, the generalized external force vector of the system is obtained. (5) Assemble the mass matrix, generalized external force vector and nodal coordinate vector of each unit according to the finite element rules to form the overall system dynamic equation, and express it in the form of a differential-algebraic equation system; (6) The system dynamic equations are solved by using the augmented method combined with numerical integration to obtain the absolute node coordinate vectors of the riser at different times, and then the tension along the entire length of the riser is calculated.
2. The method for analyzing the tension characteristics of a gently sloping riser according to claim 1, characterized in that, The wave-shaped riser system in step (1) includes a wave-shaped riser, a platform, and a seabed. The upper end of the wave-shaped riser is connected to the platform, and the lower end of the wave-shaped riser is connected to the seabed. The wave-shaped riser includes a suspended section, a buoyancy section, a descent section, and a ground-lying section. The contact point between the wave-shaped riser and the seabed is the contact point, and the riser between the contact point and the lower end is the ground-lying section. The structural parameters of the wave-shaped riser include the coordinates of the upper end and the lower end, the riser length, the riser outer diameter, the riser wall thickness, the material density, the bending stiffness, the starting position of the buoyancy section, the arrangement length of the buoyancy section, the drag coefficient, and the inertia coefficient. The marine environmental parameters include water depth, seawater density, ocean current velocity, and wave parameters. The wave parameters include wave height, period, and incident direction.
3. The method for analyzing the tension characteristics of a gently sloping riser according to claim 2, characterized in that, In step (1), the starting position and arrangement length of the buoyancy section are set according to the buoyancy ratio. The buoyancy ratio is defined as the ratio of the net buoyancy of the buoyancy section to the net underwater weight of the riser, and the value range is 10%–50%. The equivalent buoyancy distribution of the buoyancy section is calculated based on the buoyancy ratio, and the arrangement range of the buoyancy section in the riser axis is determined.
4. The method for analyzing the tension characteristics of a gently sloping riser according to claim 2, characterized in that, The buoyancy section has a length of 150-250m and its starting position is located within a range of 700-900m from the upper end point.
5. The method for analyzing the tension characteristics of a gently sloping riser according to claim 1, characterized in that, In step (2), the coordinate system of the wave riser system is as follows: the center point where the platform intersects with the sea level is the origin of the XYZ coordinate system, the sea level is the XY plane, the ocean current direction is set along the positive X-axis, and the Z-axis is perpendicular to the XY plane; the top surge response of the platform under wave action is simulated by applying simple harmonic motion along the X-axis direction at the top of the platform.
6. The method for analyzing the tension characteristics of a gently sloping riser according to claim 5, characterized in that, Step (2) specifically includes the following steps: The nodal coordinate vector of a reduced beam element consists of the global position vectors of the nodes at both ends of the element and their first-order gradient vectors with respect to the local coordinates of the element. The global position vector of any point on the element is obtained by interpolating the nodal coordinate vectors using a cubic Hermite shape function. The nodal coordinates of the reduced beam element consist of the positions of its two end nodes and their derivatives with respect to the local coordinates of the element. Under the assumption of a rigid section, the length is... The nodal coordinate vector of the reduced beam element for: ， in, To reduce the beam element The coordinate vector at the starting point of the pipeline, To reduce the beam element The coordinate vector at the end of the pipeline. for The position and direction vector at the starting point of the pipeline. for The position and direction vector at the end of the pipeline. for Tangent direction vector at the starting point of the pipe. for The tangent direction vector at the end of the pipe. for The absolute position coordinates of the pipeline starting point in the x-direction. for The absolute position coordinates of the pipeline starting point in the y-direction. for The absolute position coordinates of the pipeline starting point in the z-direction. for The absolute position coordinates of the pipeline starting point in the x-direction. for The absolute position coordinates of the pipeline starting point in the y-direction. for The absolute position coordinates of the pipeline starting point in the z-direction. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate x of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate y of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the element's axial coordinate z. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate x of the element. for The partial derivative of the position vector at the starting point of the pipe with respect to the axial coordinate y of the element. for The partial derivative of the position vector at the starting point of the pipeline with respect to the axial coordinate z of the element; For the global position vector of any point P on the axis of the reduced beam element The following is obtained through cubic Hermite interpolation of the absolute node coordinates of the element: ， in, To reduce the absolute nodal coordinate vector of the beam element, which is time-dependent; It is a three-point quadratic Hermitian function, expressed as follows: ， Where I3 is the third-order identity matrix. For dimensionless parameters, To reduce the length of the beam element; Since the absolute node coordinates of the element are only in time... The function, therefore the corresponding global speed and acceleration vector It can be represented as: ， To reduce the absolute nodal coordinate vector of beam elements The derivative of To reduce the absolute nodal coordinate vector of beam elements The second derivative; Similarly, the derivative of the position vector of a point on an element with respect to the local coordinates of the element can be expressed as: ， in, Represents global position coordinates The first partial derivative with respect to the local coordinate x, Represents the shape function matrix The first partial derivative with respect to the local coordinate x, Represents global position coordinates The second-order partial derivative with respect to the local coordinate x, Represents the shape function matrix The second partial derivative with respect to the local coordinate x; The kinetic energy expression of the element is derived from the velocity vector of a point on the element: ， in, Represented as the cross-sectional area of the element. For unit density, The unit mass matrix, The global velocity vector at a point on the element. Let be the transpose of the global velocity vector at a point on the element. Let be the velocity vector of a point on the element. Let be the transpose of the velocity vector at a point on the element. is the transpose matrix of the shape functions of the element; when When determined, The element mass matrix is a constant matrix, and its value is derived from the kinetic energy expression. : ， in, , M 21 , The unit mass matrix M is divided into four 6×6 matrices.
7. The method for analyzing the tension characteristics of a gently sloping riser according to claim 6, characterized in that, Step (3) specifically includes the following steps: Axial strain is calculated using the norm of the first derivative of the element position vector with respect to the local coordinates; bending curvature is calculated using the cross product of the first and second derivatives of the element position vector with respect to the local coordinates; and generalized elastic force is obtained by taking the partial derivative of the strain energy with respect to the element nodal coordinate vector. Strain energy of riser unit The expression is: ， Where E is the elastic modulus of the material, and J is the moment of inertia of the beam section; For axial strain, it is expressed as: ， in, Let x be the first partial derivative of the position vector with respect to the local coordinate x. Let be the transpose of the first-order partial derivative of the position vector with respect to the local coordinate x; For curvature, it is expressed as: ， Strain energy of riser unit For the coordinate vector of the unit node By taking the partial derivatives, we can obtain the generalized elastic force of the element generated by the strain energy. , is represented as: ， in, The elastic force generated by the axial deformation of the unit. The elastic force generated by the bending deformation of the unit; ， ， in For curvature absolute node coordinate vector The partial derivatives yield the transpose matrix.
8. The method for analyzing the tension characteristics of a gently sloping riser according to claim 7, characterized in that, Step (4) specifically includes the following steps: Gravity load on riser unit Represented as: ， The load of buoyancy acting on the riser unit is expressed as: ， in The density of seawater, It is the acceleration due to gravity; Hydrodynamic loads calculated based on the Morison equation include tangential drag force, normal drag force, and inertial force. The nonlinear drag in the Morison equation is divided into tangential force and normal force. Let the tangential unit vector of the pipe element be: ， For any vector a, its tangential projection onto the riser unit is: Then its normal projection is The normal projection transformation matrix is obtained. The expression is: ， In the Morison equation, the tangential drag force and normal drag force They are represented as follows: ， ， in, The diameter of the riser pipe. The density of seawater, For relative velocity, The tangential component of the relative velocity, The normal component of the relative velocity. For tangential loading function, This is the normal load function; This is the drag force coefficient; Among them, the drag coefficient Based on semi-empirical formulas and Reynolds numbers calculated using normal relative velocity and riser diameter. Related determination, represented as: ， Tangential loading function and normal loading function Represented as: ， in, The relative velocity of the ocean current; Tangential component of relative velocity and the normal component of relative velocity Represented as: ， Seabed forces are simulated using a linear spring model to represent the normal support force on the seabed. For the contact point, the linear spring support force... for: ， in, The depth of intrusion; Equivalent stiffness; Seabed depth; The unit direction vector in the Z direction; Let P be the position vector of any point; Obtain the generalized force vector The generalized force vector includes the system's generalized elastic force. and system generalized external forces , ， Among them, the generalized external force of the system The expression is: ， in The load on the riser unit is caused by gravity. buoyancy load on riser unit Tangential drag force Normal drag force and linear spring support force constitute.
9. The method for analyzing the tension characteristics of a gently sloping riser according to claim 8, characterized in that, Step (5) specifically includes the following steps: The system of dynamic differential algebraic equations for the constrained system is established using the Lagrange multiplier method, and its form is as follows: The dynamic equations of the ANCF mooring cable are a set of index-3 differential-algebraic equations. Solving them using the Baumgarte method and rewriting the formulas in matrix form yields a set of index-1 differential-algebraic equations. ， in, The system's quality matrix, For constraint equations For generalized coordinates Jacobi matrix, for The transpose of the matrix, Let be the system acceleration vector. For Lagrange multipliers, It is a generalized force vector. The right-hand side of the constraint acceleration equation includes velocity nonlinear terms and explicit time derivative terms; The system constraint equations The first and second partial derivatives with respect to time are as follows: ， ， in, To constrain the Jacobian matrix for generalized coordinates, To constrain the explicit partial derivatives with respect to time, For velocity vectors, For acceleration vectors, To constrain the second-order explicit partial derivatives with respect to time, for right The partial derivatives, These are mixed partial derivatives, i.e., constraint pairs. Again The partial derivative; The right-hand side of the constrained acceleration equation is obtained from the above equation. : ， In the formula: ， We obtain a set of ordinary differential equations, namely the system dynamics equations: 。 10. The method for analyzing the tension characteristics of a gently sloping riser according to claim 9, characterized in that, Step (6) specifically includes the following steps: The augmented method and the introduction of the Baumgarte default stabilization term are used to numerically solve the dynamic differential-algebraic equation system, transforming the differential-algebraic equation system of index -3 into a system of first-order ordinary differential equations that can be directly integrated. To address potential breaches of the equations during numerical computation, a breach correction method based on control feedback is adopted, which involves introducing a feedback system into the constraint equations. ， in, The right-hand side of the constraint equations after the Baumgarte method correction is represented. , and are stability coefficients. Empirical values for and are between 5 and 50. When = , the stable response is fast, resulting in a system of first-order differential equations: ， In the formula, B represents the first N on the right side. dof Line, i.e., the system acceleration vector, N dof Let be the total number of degrees of freedom of the system; therefore, the system's velocity and position vectors can be obtained by solving the system's differential equations; By solving the matrix-form equations of motion dynamically, the absolute nodal coordinate vectors of the riser unit are obtained. ,pass The position vector of the riser unit during the simulation process can be calculated. Elastic force The elastic force generated by the axial deformation of the riser unit This refers to the tension of the riser.