Non-linear dynamic fast integration algorithm for large-flexibility mooring anchor chain

Abstract

The invention belongs to the field of computational mechanics, and discloses a nonlinear dynamic fast integration algorithm for a large-flexibility mooring anchor chain. An efficient algorithm for the nonlinear dynamics problem of the mooring anchor chain is established on the basis of a fine integration method and a prediction correction algorithm, the tension and compression nonlinearity of the anchor chain is considered, linear processing is conducted on a stiffness matrix of the equation, meanwhile, integral analysis is conducted on linear external force, polynomial interpolation approximation is conducted on the nonlinear external force, and the nonlinear dynamic problem of the mooring anchor chain is solved. And the calculation precision and efficiency of the algorithm are obviously improved. The provided efficient numerical algorithm can more quickly and accurately simulate the mechanical property and the structural response of the mooring anchor chain, and can more accurately predict the impact load and the movement trend of the anchor chain, so that the damage condition of the anchor chain is judged, the reasonable design of parameter indexes such as the strength of the anchor chain is helped, and the method has important engineering significance.

Description

Technical Field

[0001] The invention belongs to the field of computational mechanics and relates to a fast integration algorithm for nonlinear dynamics of a large-flexibility mooring anchor chain. Background Art

[0002] The mooring system of the deep-sea platform is a complex, time-varying dynamic system that is subject to the combined effects of complex loads such as wind, waves, and currents. In addition, there is a strong nonlinear effect between the floating body and the mooring system. When the platform has large displacements and large rotations, the geometric nonlinearity of the mooring chain and the nonlinearity of the chain-seabed contact need to be considered. Traditional methods often use means such as applying pre-tension to avoid nonlinear behavior, but the deep-sea conditions are complex and the nonlinear behavior of the chain is inevitable. Therefore, it is of great significance to carry out research on efficient simulation technology for the strong nonlinear behavior of large-flexibility mooring chains and establish an efficient nonlinear dynamic simulation algorithm for mooring chains.

[0003] Current methods, such as Tang Yougang's "Calculation of deep-sea mooring impact tension by lumped mass method" and Hall's "Validation of a lumped-mass mooring line model with DeepCwindsemisubmersible model testdata", mostly use the lumped mass method to assume that the anchor chain is a nonlinear catenary composed of many lumped mass points and massless springs, and the gravity, tension, seabed contact force, damping force, additional mass force and fluid drag force are equivalent to the nodes to establish the dynamic calculation model of the anchor chain. For the treatment of the nonlinearity of the mooring anchor chain and the nonlinearity of the anchor chain-seabed contact, the traditional method mostly uses the second-order Runge-Kutta method with a constant time step to solve it. According to the tension-relaxation state of the anchor chain at the current moment and the contact with the seabed, the tension of the anchor chain and the seabed contact force are calculated, and the dynamic integration is performed. Since the mooring system is a complex nonlinear system, the motion state at the current moment cannot well predict the motion state of the anchor chain at the next moment, so this simplified calculation method will affect the simulation accuracy of the mooring system.

[0004] In order to solve the above technical problems, the present invention establishes an efficient algorithm for solving the nonlinear dynamics problem of mooring anchor chain based on the precise integration method and the prediction-correction algorithm. The nonlinear behavior of the equation is updated at each time step and solved iteratively. In view of the geometric nonlinear behavior of the anchor chain, the stiffness matrix of the equation is linearized, and the linear part and the nonlinear part of the equation are separated. The linear external force is solved by analytical integration, and the nonlinear external force is approximated by polynomial interpolation, thereby significantly improving the efficiency and accuracy of the nonlinear dynamic simulation of the mooring anchor chain. Summary of the invention

[0005] The purpose of the invention is to provide a fast integration algorithm for nonlinear dynamics of a large-flexibility mooring chain.

[0006] The technical solution of the present invention:

[0007] A fast integration algorithm for nonlinear dynamics of high-flexibility mooring chains, the steps are as follows:

[0008] (1) Establish the nonlinear dynamic equation of the mooring chain

[0009] The nonlinear dynamic equation of the mooring chain is established based on the lumped mass method;

[0010] First, the mooring chain is equivalent to a nonlinear catenary composed of many concentrated mass points and massless springs. The points where the concentrated mass is located are called nodes. Then, the tension, internal damping force, gravity, seabed contact force, additional mass force and hydrodynamic resistance of the mooring chain are equivalently applied to the nodes, and the node motion differential equation of the mooring chain is established:

[0011]

[0012] In the formula, m i is the mass of the node, m ai is the additional mass of the node, is the node acceleration, T i+(1 / 2) and C i+(1 / 2) is the tension and internal damping force acting on segment i+1 / 2, D qi and D pi are the tangential and normal drag forces acting on the node, W i and B i is the gravity and buoyancy of the node;

[0013] Then all nodes are assembled in a finite element manner to construct the corresponding mass matrix, stiffness matrix, damping matrix and external force terms, and the nonlinear dynamic equation of the mooring chain is obtained as follows:

[0014]

[0015] In the formula, M is the mass matrix, C is the damping matrix, K is the stiffness matrix, g is the external force term, r, and are the displacement, velocity and acceleration of the anchor chain respectively; the motion state of each node of the mooring anchor chain at the initial moment is taken as the initial condition, and the motion state of the fixed end and the excitation end of the mooring anchor chain at any moment is taken as the boundary condition, on this basis, the nonlinear dynamic equation of the mooring anchor chain is numerically solved;

[0016]

[0017] In the formula, r(0) represents tk = the initial state vector at time 0, r0(t) represents the state vector of the anchor chain fixed end at any time, r N (t) represents the state vector of the anchor chain excitation end at any time, and is the corresponding derivative expression;

[0018] (2) Linearization processing

[0019] The nonlinear dynamic equation of the mooring chain is linearized, that is, the linear part and the nonlinear part in the nonlinear dynamic equation of the mooring chain are separated; the mass matrix, damping matrix and external force items of the mooring chain are extracted as linear matrices according to the nonlinear dynamic equation of the mooring chain; the linear external force and the nonlinear external force of the mooring chain are separated, so that the nonlinear dynamic equation of the mooring chain is written as:

[0020]

[0021] Where ω is the mooring chain excitation frequency, g 00 , g s and g c are constant external forces, g1(t,r) is a nonlinear external force; for the mass matrix and stiffness matrix, t k = 0 is approximately the corresponding linear part, that is:

[0022] M L =M(0),C L =C(0) (5)

[0023] Where M L and C L are the linear mass matrix and the linear stiffness matrix, M(0) and C(0) are t k = 0; however, since the mooring chain only has tension when stretched and is zero when relaxed, there is a nonlinear tension-compression characteristic, so the stiffness matrix needs to be linearized; the tension expression of any node i in the local coordinate system is known to be:

[0024] T i =K i Δu i (6)

[0025] In the formula, Δu i =u i+1 -u i is the relative displacement between two adjacent nodes, K i is the linear stiffness coefficient, which has a piecewise characteristic and is expressed as follows:

[0026]

[0027] Where E is the elastic modulus of the mooring chain, d is the equivalent diameter, and l is the unstretched length. i =[x i y i z i ] T , considering the coordinate change relationship between the local coordinates and the global coordinates, we get:

[0028]

[0029] Where θ and γ are the angles between the segmented axial direction of the mooring chain and the x-axis and z-axis directions of the global coordinate system, respectively. The expressions are as follows:

[0030]

[0031] Therefore, there is a coordinate transformation relationship:

[0032] Δu i =Θ i {r i r i+1} T (10)

[0033] In the formula, r i and r i+1 are the displacements of nodes i and i+1, Θ i is the coordinate transformation vector, and its specific expression is as follows:

[0034] Θ i =[-cos(θ) -sin(θ) -cos(γ) cos(θ) sin(θ) cos(γ)] (11)

[0035] Knowing the relationship between the local coordinates and the overall coordinates of the mooring chain, all segments of the mooring chain are assembled in a finite element manner to obtain the overall linear stiffness matrix:

[0036]

[0037] Where N e Indicates the number of anchor chain segments;

[0038] (3) Construct an efficient integration algorithm for solving the nonlinear dynamics of mooring chains

[0039] Based on the precise integration method, the linearization of the mooring chain is considered at the same time. The linear external force is solved analytically, and the nonlinear external force is solved approximately by polynomial interpolation. An efficient integration algorithm for the nonlinear dynamics of the mooring chain is constructed. First, the state vector is introduced The nonlinear dynamic equation of the mooring anchor chain is reduced from the second-order differential equation to the first-order, and the dynamic equation is written as:

[0040]

[0041] Where A is a linear constant matrix, which is composed of the linear mass matrix M L , linear stiffness matrix K L And the linear damping array C L are obtained by linearization; f0, f s 、f c is the linear external force term, and f(t,v) is the nonlinear external force term. The specific expressions are as follows:

[0042]

[0043] The nonlinear dynamic equation of the mooring chain in the section [t k ,t k+1 The integral solution on ] is:

[0044]

[0045] Where, T = exp(Aη), η = t k+1 -t k ,τ∈(0,η], For the linear external force term, the corresponding analytical expression is obtained by direct integration:

[0046]

[0047] in

[0048]

[0049] Where I is the unit matrix; for nonlinear external force terms, Lagrange polynomial expansion is used; if the interpolation point does not contain t k+1 The state at the moment, the explicit format is obtained, otherwise, the implicit format is obtained; the following is the derivation process of the implicit two-step method based on precise integration, selecting t k-1 ,t k and t k+1 The external forces at three moments are used as interpolation points.

[0050] f(t k +τ,v)=l k-1 f k-1 +l k f k +l k+1 f k+1 (18)

[0051] In the formula, f k-1 、fk and f k+1 t k-1 ,t k and t k+1 The nonlinear external force at time l k-1 , l k and l k+1 is the corresponding Lagrange interpolation basis function, and the unified expression is:

[0052]

[0053] Substituting into equation (16) and integrating it, the implicit recursive expression of the nonlinear dynamics of the mooring chain is:

[0054]

[0055] The corresponding explicit recursive expression is:

[0056]

[0057] in

[0058]

[0059] For the explicit integration algorithm, the motion response at the next moment is obtained directly by recursion, while when the implicit algorithm is solved, due to t k+1 The state at the moment is unknown, so the prediction-correction model needs to be used to iteratively solve; at the initial moment, let v k+1 =v k , then the corresponding nonlinear external force is f k+1 =f(v k+1 ,t k+1 ), substitute into the recursive expression, and we get t k+1 The state vector at time The newly obtained displacement The displacement v assumed at the initial moment k+1 In comparison, if the relative error is less than 1e-8, it is considered that the accuracy requirement is met and the iterative solution is correct; otherwise, the state vector obtained by recursion is used as the initial condition, that is, Repeat the above iterative process until the accuracy requirement is met;

[0060] (4) Solve the Duhamel integral matrix

[0061] For the integral matrix T in the above recursive expression mThe solution is solved by the precise integration method, using the series expansion characteristics and addition characteristics of the matrix exponent; first, the matrix exponent is expanded in series, and the series expansion is substituted into the expression of the integral matrix, and the series expression of the integral matrix is ​​obtained by integration; since the increment matrix in the expression is very small, the constant matrix and the increment matrix need to be stored separately, and the increment matrix needs to be calculated separately; then, the segment is divided, and the small segment τ=η / 2 is taken N , according to the series expansion, the incremental matrix on the small segment is obtained, and the addition characteristics of the matrix exponent are used to substitute into the matrix addition formula, and it is repeated N times to obtain the incremental matrix of the matrix. Adding the constant matrix gives the Duhamel integral matrix.

[0062] Beneficial effects of the present invention: In view of the problems of low computational efficiency, low precision and small desirable computational time step of traditional algorithms, the present invention establishes an efficient algorithm for the nonlinear dynamics problem of mooring anchor chains based on the fine integration method and the prediction and correction algorithm, and linearizes the stiffness matrix of the equation by considering the tension and compression nonlinearity of the anchor chain. At the same time, the linear external force is analytically integrated, and the nonlinear external force is approximated by polynomial interpolation, which significantly improves the computational accuracy and efficiency of the algorithm. The proposed efficient numerical algorithm can simulate the mechanical properties and structural response of the mooring anchor chain more quickly and accurately, and can more accurately predict the impact load size and movement trend of the anchor chain, thereby judging the damage conditions of the anchor chain, and helping to reasonably design parameter indicators such as the anchor chain strength, which has important engineering significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 It is the algorithm flow chart.

[0064] Figure 2 It is the force analysis of anchor chain node.

[0065] Figure 3 It is a transformation diagram between local coordinates and global coordinates.

[0066] Figure 4 It is a schematic diagram of the implicit two-step interpolation points.

[0067] Figure 5 It is the solution flow chart of implicit integration algorithm.

[0068] Figure 6 It is a simulation diagram of mooring anchor chain.

[0069] Figure 7 It is the time history of node motion response when the step size is 0.001s, where (a) is the x-direction displacement of 6 nodes and (b) is the z-direction displacement of 12 nodes.

[0070] Figure 8 is the time history of tension and seabed contact force at a step size of 0.001 s, where (a) is the tension of the 20th segment and (b) is the seabed contact force at 1 node.

[0071] Fig. 9 It is the time history of node motion response when the step size is 0.006s, where (a) is the x-direction displacement of 6 nodes and (b) is the z-direction displacement of 12 nodes.

[0072] Fig.10 is the time history of tension and seabed contact force at a step size of 0.006 s, where (a) is the tension at the 20th segment and (b) is the seabed contact force at 1 node.

[0073] Fig.11 It is the time history of node motion response when the step size is 0.06s, where (a) is the x-direction displacement of 6 nodes and (b) is the z-direction displacement of 12 nodes.

[0074] Fig.12 is the time history of tension and seabed contact force at a step size of 0.06 s, where (a) is the tension at the 20th segment and (b) is the seabed contact force at 1 node. DETAILED DESCRIPTION

[0075] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0076] Based on the precise integration method and prediction-correction algorithm, an efficient integration algorithm for the nonlinear dynamics of mooring chains was established, and efficient simulation of large-flexibility mooring chains was achieved. The specific technical solutions are as follows:

[0077] (1) Establish the nonlinear dynamic equation of the mooring chain

[0078] The nonlinear dynamic equation of the mooring chain is established based on the concentrated mass method. First, the anchor chain is equivalent to a nonlinear catenary composed of many concentrated mass points and massless springs. The points where the concentrated mass is located are called nodes. Then, the tension, internal damping force, gravity, seabed contact force, additional mass force and hydrodynamic resistance of the anchor chain are equivalently applied to the nodes, and the node motion differential equation of the mooring chain is established:

[0079]

[0080] In the formula, m i is the mass of the node, m ai is the additional mass of the node, r i is the state vector of the node, T i+(1 / 2) and C i+(1 / 2) is the tension and internal damping force acting on segment i+1 / 2, D qi and D pi are the tangential and normal drag forces acting on the node, W i and B iis the gravity and buoyancy of the node. Finally, all nodes are assembled in a finite element manner to construct the corresponding mass matrix, stiffness matrix, damping matrix and external force term, and the motion differential equation of the entire mooring anchor chain is obtained as follows:

[0081]

[0082] The motion state of each node of the anchor chain at the initial moment is taken as the initial condition, and the motion state of the fixed end and the excitation end of the anchor chain at any moment is taken as the boundary condition. On this basis, the nonlinear dynamic equation of the mooring anchor chain can be numerically solved.

[0083]

[0084] In the formula, r(0) represents t k = the initial state vector at time 0, r0(t) represents the state vector of the anchor chain fixed end at any time, r N (t) represents the state vector of the anchor chain excitation end at any time, and is the corresponding derivative expression;

[0085] (2) Linearization processing

[0086] The nonlinear dynamic equation of the mooring chain is linearized, that is, the linear part and the nonlinear part in the nonlinear dynamic equation of the mooring chain are separated; the mass matrix, damping matrix and external force items of the mooring chain are extracted as linear matrices according to the nonlinear dynamic equation of the mooring chain; the linear external force and the nonlinear external force of the mooring chain are separated, so that the nonlinear dynamic equation of the mooring chain is written as:

[0087]

[0088] Where ω is the mooring chain excitation frequency, g 00 , g s and g c are constant external forces, g1(t,r) is a nonlinear external force; for the mass matrix and stiffness matrix, t k = 0 is approximately the corresponding linear part, that is:

[0089] M L =M(0),C L =C(0) (27)

[0090] Where M L and C L are the linear mass matrix and the linear stiffness matrix, M(0) and C(0) are t k= 0. However, since the mooring chain only has tension when it is stretched and the tension is zero when it is relaxed, there are tensile and compressive nonlinear characteristics, so the stiffness matrix needs to be linearized; the tension expression of any node i in the local coordinate system is known to be:

[0091] T i =K i Δu i (28)

[0092] In the formula, Δu i =u i+1 -u i is the relative displacement between two adjacent nodes, K i is the linear stiffness coefficient, which has a piecewise characteristic and is expressed as follows:

[0093]

[0094] Where E is the elastic modulus of the mooring chain, d is the equivalent diameter, and l is the unstretched length. i =[x i y i z i ] T , considering the coordinate change relationship between the local coordinates and the global coordinates, we get:

[0095]

[0096] Where θ and γ are the angles between the segmented axial direction of the mooring chain and the x-axis and z-axis directions of the global coordinate system, respectively. The expressions are as follows:

[0097]

[0098] Therefore, there is a coordinate transformation relationship:

[0099] Δu i =Θ i {r i r i+1} T (32)

[0100] In the formula, r i and r i+1 are the displacements of nodes i and i+1, Θ i is the coordinate transformation vector, and its specific expression is as follows:

[0101] Θ i =[-cos(θ) -sin(θ) -cos(γ) cos(θ) sin(θ) cos(γ)] (33)

[0102] Knowing the relationship between the local coordinates and the overall coordinates of the mooring chain, all segments of the mooring chain are assembled in a finite element manner to obtain the overall linear stiffness matrix:

[0103]

[0104] Where N e Indicates the number of anchor chain segments;

[0105] (3) Construct an efficient integration algorithm for solving the nonlinear dynamics of mooring chains

[0106] Based on the precise integration method, the linearization of the mooring chain is considered at the same time. The linear external force is solved analytically, and the nonlinear external force is solved approximately by polynomial interpolation. An efficient integration algorithm for the nonlinear dynamics of the mooring chain is constructed. First, the state vector is introduced The nonlinear dynamic equation of the mooring anchor chain is reduced from the second-order differential equation to the first-order, and the dynamic equation is written as:

[0107]

[0108] Where A is a linear constant matrix, which is composed of the linear mass matrix M L , linear stiffness matrix K L And the linear damping array C L are obtained by linearization; f0, f s 、f c is the linear external force term, and f(t,v) is the nonlinear external force term. The specific expressions are as follows:

[0109]

[0110] The nonlinear dynamic equation of the mooring chain in the section [t k ,t k+1 The integral solution on ] is:

[0111]

[0112] Where, T = exp(Aη), η = t k+1 -t k ,τ∈(0,η], For the linear external force term, the corresponding analytical expression is obtained by direct integration:

[0113]

[0114] in

[0115]

[0116] Where I is the unit matrix; for nonlinear external force terms, Lagrange polynomial expansion is used; if the interpolation point does not contain t k+1 The state at the moment, the explicit format is obtained, otherwise, the implicit format is obtained; the following is the derivation process of the implicit two-step method based on precise integration, selecting t k-1 ,t k and t k+1 The external forces at three moments are used as interpolation points.

[0117] f(t k +τ,v)=l k-1 f k-1 +l k f k +l k+1 f k+1 (40)

[0118] In the formula, f k-1 、f k and f k+1 t k-1 ,t k and t k+1 The nonlinear external force at time l k-1 , l k and l k+1 is the corresponding Lagrange interpolation basis function, and the unified expression is:

[0119]

[0120] Substituting into equation (38) and integrating it, the implicit recursive expression of the nonlinear dynamics of the mooring chain is:

[0121]

[0122] The corresponding explicit recursive expression is:

[0123]

[0124] in

[0125]

[0126] For the explicit integration algorithm, the motion response at the next moment is obtained directly by recursion, while when the implicit algorithm is solved, due to t k+1 The state at the moment is unknown, so the prediction-correction model needs to be used to iteratively solve; at the initial moment, let v k+1 =v k , then the corresponding nonlinear external force is f k+1 =f(v k+1 ,t k+1 ), substitute into the recursive expression, and we get t k+1 The state vector at time The newly obtained displacement The displacement v assumed at the initial moment k+1 In comparison, if the relative error is less than 1e-8, it is considered that the accuracy requirement is met and the iterative solution is correct; otherwise, the state vector obtained by recursion is used as the initial condition, that is, Repeat the above iterative process until the accuracy requirement is met;

[0127] (4) Solve the Duhamel integral matrix

[0128] For the integral matrix T in the above recursive expression m The solution can be solved by the precise integration method, making full use of the series expansion characteristics and addition characteristics of the matrix exponent. First, the matrix exponent is expanded in series, and the series expansion is substituted into the expression of the integral matrix. The integral matrix is ​​integrated to obtain the series expression of the integral matrix. Since the increment matrix in the expression is very small, the constant matrix and the increment matrix need to be stored separately, and the increment matrix needs to be calculated separately. Then, the segment is divided, and a very small segment τ = η / 2 is taken N , according to the series expansion, the incremental matrix on the small segment can be obtained. By using the addition characteristics of the matrix exponent, substituting it into the addition formula of the matrix and looping N times, the incremental matrix of the matrix can be obtained. Adding the constant matrix can get the Duhamel integral matrix. The precise integration method is applied to the precise solution of the matrix, which greatly improves the calculation accuracy and efficiency of the solution;

[0129] (5) Numerical simulation

[0130] The OC3-Hywind 5MW Spar floating wind turbine is used as the simulation object to simulate the dynamic behavior of the wind turbine platform, and the simulation results of the implicit two-step method constructed based on fine integration in this paper are compared with the simulation results of the existing open source software MoorDyn. The displacement excitation of the upper end point of the anchor chain is taken as a horizontal sinusoidal excitation z(t) = 10sin(1.1t). Since the states of the seabed fixed end and the excitation end are known, it is only necessary to solve the motion state of each point in the middle. The anchor chain is divided into twenty sections and solved using the implicit two-step method and MoorDyn software respectively. The results are plotted at intervals of 0.48s;

[0131] The result obtained by MoorDyn software with a small step size of 0.001s is used as the reference solution. When the calculation time step is 0.001s, the results of the implicit two-step method are consistent with the results of MoorDyn software in terms of displacement, tension and seabed contact force, which verifies the correctness of the algorithm in this paper when considering the geometric nonlinearity of the anchor chain and the nonlinearity of the anchor chain-seabed contact. Further increasing the calculation time step, it can be seen from the calculation that the maximum time step that MoorDyn software can solve is 0.006s. When the large step size is 0.006s, the result of the implicit two-step method is consistent with the reference solution, while the seabed contact force of MoorDyn software has shown obvious deviations. This shows that under the same solution time step, the solution accuracy of the algorithm in this paper is higher and the stability of the algorithm is better. For the implicit two-step method, the calculation time step can be further increased to 0.06s. When the large step size is 0.06s, the displacement and tension results of the implicit two-step method are consistent with the reference solution. This shows that the algorithm in this paper can solve a larger time step and has better stability;

[0132] In summary, the maximum calculation time step that the algorithm in this paper can take is about ten times that of the open source software MoorDyn, and the algorithm has better stability. After obtaining the motion response of the mooring anchor chain, the force conditions of the mooring anchor chain can be further calculated and analyzed, such as nonlinear tension and seabed contact force, and then the strength failure conditions of the anchor chain can be determined. By simulating the movement of the mooring anchor chain under different excitations, better design, reasonable layout, and selection of more appropriate anchor chain strength, number of anchor chains, angle between anchor chains, etc., the stability of the mooring anchor chain system can be improved.

Claims

1. A fast integration algorithm for nonlinear dynamics of high-flexibility mooring chains, characterized in that: Here are the steps: (1) Establish the nonlinear dynamic equation of the mooring chain The nonlinear dynamic equation of the mooring chain is established based on the lumped mass method; First, the mooring chain is equivalent to a nonlinear catenary composed of many concentrated mass points and massless springs. The points where the concentrated mass is located are called nodes. Then, the tension, internal damping force, gravity, seabed contact force, additional mass force and hydrodynamic resistance of the mooring chain are equivalently applied to the nodes, and the node motion differential equation of the mooring chain is established: In the formula, m i is the mass of the node, m ai is the additional mass of the node, is the node acceleration, T i+(1 / 2) and C i+(1 / 2) is the tension and internal damping force acting on segment i+1 / 2, D qi and D pi are the tangential and normal drag forces acting on the node, W i and B i is the gravity and buoyancy of the node; Then all nodes are assembled in a finite element manner to construct the corresponding mass matrix, stiffness matrix, damping matrix and external force terms, and the nonlinear dynamic equation of the mooring chain is obtained as follows: In the formula, M is the mass matrix, C is the damping matrix, K is the stiffness matrix, g is the external force term, r, and are the displacement, velocity and acceleration of the anchor chain respectively; the motion state of each node of the mooring anchor chain at the initial moment is taken as the initial condition, and the motion state of the fixed end and the excitation end of the mooring anchor chain at any moment is taken as the boundary condition, on this basis, the nonlinear dynamic equation of the mooring anchor chain is numerically solved; In the formula, r(0) represents t k = the initial state vector at time 0, r0(t) represents the state vector of the anchor chain fixed end at any time, r N (t) represents the state vector of the anchor chain excitation end at any time, and is the corresponding derivative expression; (2) Linearization processing The nonlinear dynamic equation of the mooring anchor chain is linearized, that is, the linear part and the nonlinear part of the nonlinear dynamic equation of the mooring anchor chain are separated; The mass matrix, damping matrix and external force of the mooring chain are extracted according to the nonlinear dynamic equation of the mooring chain. The linear part is used as a linear matrix; the linear external force and nonlinear external force of the mooring chain are separated, so that the nonlinear dynamic equation of the mooring chain is written as: Where ω is the mooring chain excitation frequency, g 00 , g s and g c are constant external forces, g1(t,r) is a nonlinear external force; for the mass matrix and stiffness matrix, t k = 0 is approximately the corresponding linear part, that is: M L =M(0),C L =C(0) (5) Where M L and C L are the linear mass matrix and the linear stiffness matrix, M(0) and C(0) are t k = 0; however, since the mooring chain only has tension when stretched and is zero when relaxed, there is a nonlinear tension-compression characteristic, so the stiffness matrix needs to be linearized; the tension expression of any node i in the local coordinate system is known to be: T i =K i Δu i (6) In the formula, Δu i =u i+1 -u i is the relative displacement between two adjacent nodes, K i is the linear stiffness coefficient, which has a piecewise characteristic and is expressed as follows: Where E is the elastic modulus of the mooring chain, d is the equivalent diameter, and l is the unstretched length. i =[x i y i z i ] T , considering the coordinate change relationship between the local coordinates and the global coordinates, we get: Where θ and γ are the angles between the segmented axial direction of the mooring chain and the x-axis and z-axis directions of the global coordinate system, respectively. The expressions are as follows: Therefore, there is a coordinate transformation relationship: Thu i =Θ i {r} i r i+1 } T (10) In the formula, r i and r i+1 are the displacements of nodes i and i+1, Θ i is the coordinate transformation vector, and its specific expression is as follows: I i =[-cos(θ)-sin(θ)-cos(γ)cos(θ)sin(θ)cos(γ)] (11) Knowing the relationship between the local coordinates and the overall coordinates of the mooring chain, all segments of the mooring chain are assembled in a finite element manner to obtain the overall linear stiffness matrix: Where N e Indicates the number of anchor chain segments; (3) Construct an efficient integration algorithm for solving the nonlinear dynamics of mooring chains Based on the precise integration method, the linearization of the mooring chain is considered at the same time. The linear external force is solved analytically, and the nonlinear external force is solved approximately by polynomial interpolation. An efficient integration algorithm for the nonlinear dynamics of the mooring chain is constructed. First, the state vector is introduced The nonlinear dynamic equation of the mooring anchor chain is reduced from the second-order differential equation to the first-order, and the dynamic equation is written as: Where A is a linear constant matrix, which is composed of the linear mass matrix M L , linear stiffness matrix K L And the linear damping array C L are obtained by linearization; f0, f s 、f c is the linear external force term, and f(t,v) is the nonlinear external force term. The specific expressions are as follows: The nonlinear dynamic equation of the mooring chain in the section [t k ,t k+1 The integral solution on ] is: Where, T = exp(Aη), η = t k+1 -t k ,τ∈(0,η], For the linear external force term, the corresponding analytical expression is obtained by direct integration: in Where I is the unit matrix; for nonlinear external force terms, Lagrange polynomial expansion is used; if the interpolation point does not contain t k+1 The state at the moment, the explicit format is obtained, otherwise, the implicit format is obtained; the following is the derivation process of the implicit two-step method based on precise integration, selecting t k-1 ,t k and t k+1 The external forces at three moments are used as interpolation points. f(t k +τ,v)=l k-1 f k-1 +l k f k +l k+1 f k+1 (18) In the formula, f k-1 、f k and f k+1 t k-1 ,t k and t k+1 The nonlinear external force at time l k-1 , l k and l k+1 is the corresponding Lagrange interpolation basis function, and the unified expression is: Substituting into equation (16) and integrating it, the implicit recursive expression of the nonlinear dynamics of the mooring chain is: The corresponding explicit recursive expression is: in For the explicit integration algorithm, the motion response at the next moment is obtained directly by recursion, while when the implicit algorithm is solved, due to t k+1 The state at the moment is unknown, so the prediction-correction model needs to be used to iteratively solve; at the initial moment, let v k+1 =v k , then the corresponding nonlinear external force is f k+1 =f(v k+1 ,t k+1 ), substitute into the recursive expression, and we get tk +1 The state vector at time The newly obtained displacement The displacement v assumed at the initial moment k+1 In comparison, if the relative error is less than 1e-8, it is considered that the accuracy requirement is met and the iterative solution is correct; otherwise, the state vector obtained by recursion is used as the initial condition, that is, Repeat the above iterative process until the accuracy requirement is met; (4) Solve the Duhamel integral matrix For the integral matrix T in the above recursive expression m The solution is solved by the precise integration method, using the series expansion characteristics and addition characteristics of the matrix exponent; first, the matrix exponent is expanded in series, and the series expansion is substituted into the expression of the integral matrix, and the series expression of the integral matrix is ​​obtained by integration; since the increment matrix in the expression is very small, the constant matrix and the increment matrix need to be stored separately, and the increment matrix needs to be calculated separately; then, the segment is divided, and the small segment τ=η / 2 is taken N , according to the series expansion, the incremental matrix on the small segment is obtained, and the addition characteristics of the matrix exponent are used to substitute into the matrix addition formula, and it is repeated N times to obtain the incremental matrix of the matrix. Adding the constant matrix gives the Duhamel integral matrix.

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