A method for calculating high-frequency resonance response of waves and tall structures and a computer-readable storage medium

Abstract

A method for calculating the high-frequency resonance response of waves and tall structures includes obtaining dimensional information of the tall structure and calculating wave information within the water area; performing a third-order perturbation expansion on the translational component, rotational component, velocity potential, and wave surface rise; establishing a nonlinear time-domain mathematical model of the interaction between waves and the tall structure based on the third-order perturbation expansion and the obtained dimensional and wave information; obtaining the velocity potential using the nonlinear time-domain mathematical model; and then using the velocity potential to obtain the wave force and motion response of the high-frequency resonance response of waves and tall structures. The advantages of the present invention are that the use of the nonlinear time-domain mathematical model not only enables the study of the high-frequency resonance response of waves and tall structures, but also improves the computational efficiency of model testing; the nonlinear time-domain mathematical model also takes into account the effects of nonlinearity, resulting in higher accuracy in the calculation results of the high-frequency resonance response of waves and tall structures.

Description

Technical Field

[0001] The present invention relates to the field of marine engineering technology, and in particular to a method for calculating high-frequency resonance response between waves and tall structures and a computer-readable storage medium. Background Art

[0002] As offshore wind turbines generate increasingly powerful power, their towers are becoming taller. Offshore wind turbines are tall structures. Due to their inherent length and flexibility, these structures can experience a pronounced "ringing" phenomenon when subjected to wave action. Ringing refers to a near-transient, high-frequency resonant response of offshore structures under harsh sea conditions. The frequency of this motion response is significantly higher than the characteristic frequency of the waves, typically 3 to 5 times that frequency. This high-frequency resonant response increases the ultimate response amplitude of tall structures and can cause fatigue failure. Therefore, research on the high-frequency resonant response of waves and tall structures is necessary.

[0003] Currently, the calculation method for the high-frequency resonant response of waves and tall structures is shown in the Chinese invention "Method for Obtaining High-Frequency Dynamic Response of Offshore Floating Wind Turbines" with patent number CN202311091580.5 (authorization announcement number CN116816620B). This method determines the natural frequency and natural period of the floating wind turbine superstructure in the floating body motion state through physical model experiments, uses the natural period to design a regular wave test, calculates the high-frequency dynamic response caused by high-order wave effects under the corresponding period, and finally determines the retained order of high-order wave effects, thereby avoiding the neglect and excessive consideration of key high-order wave effects.

[0004] While the aforementioned method can calculate the high-frequency resonant response of waves and tall structures, it primarily relies on physical models for the calculations. This requires scaling the tall structures to a certain scale and conducting simulation tests in wave tanks or pools. This results in relatively high experimental costs and low computational efficiency. Currently, there is a lack of mathematical models for studying the high-frequency resonant response of waves and tall structures. Traditional mathematical models can only calculate the first- and second-order terms of the equation of motion, but not the third-order terms of the equation of motion for high-frequency resonant response. Therefore, the existing technology needs further improvement. Summary of the Invention

[0005] The first technical problem to be solved by the present invention is to provide a calculation method capable of quickly calculating the interaction between waves and tall structures in response to the above-mentioned existing technical status.

[0006] The second technical problem to be solved by the present invention is to provide a computer-readable storage medium for storing a calculation method capable of quickly calculating the interaction between waves and tall structures in response to the above-mentioned existing technical status.

[0007] The technical solution adopted by the present invention to solve the first technical problem is: the method for calculating the high-frequency resonance response of waves and tall structures is characterized in that the method comprises the following steps:

[0008] S1. Obtaining the size information of tall structures and calculating the wave information in the water area;

[0009] S2. Perform a third-order perturbation expansion on the translational and rotational components of the tall structure, as well as the velocity potential and wave surface rise in the calculation water area. Based on the third-order perturbation expansion and the size and wave information obtained in S1, set the boundary conditions of the calculation water area to establish a nonlinear time-domain mathematical model for studying the interaction between waves and tall structures. The nonlinear time-domain mathematical model includes the Laplace equation, the Bernoulli equation for unsteady flow, and the motion equation derived from the conservation of momentum theorem.

[0010] S3. According to the boundary conditions set in S2, the velocity potential of the fluid motion velocity distribution and change is calculated using the Laplace equation in the nonlinear time domain mathematical model;

[0011] S4. Based on the velocity potential determined in S3, the non-steady Bernoulli equation in the nonlinear time-domain mathematical model is used to calculate the pressure distribution on the surface of the tall structure. The pressure distribution is divided according to different orders to obtain the wave force of the high-frequency resonance response between the wave and the tall structure.

[0012] S5, the nonlinear time-domain mathematical model inputs the wave force obtained in S4 into the motion equation, uses the numerical integration method to solve the motion equation, and predicts the motion response of waves and tall structures in the calculated water area in real time; the motion equation includes the first-order motion equation, the second-order motion equation, and the third-order motion equation. The first-order motion equation is:

[0013]

[0014] Where, is the first-order dynamic force in the wave force, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the towering structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (1) is the first-order component of the translational component, is the time derivative of the first-order component of the translational component, that is, the first-order component of the translational velocity, is the time derivative of the first-order component of the translational velocity, i.e., the first-order component of the translational acceleration, and {} is a 6×1 matrix;

[0015] The second-order equation of motion is:

[0016]

[0017] Where, is the second-order dynamic force in the wave force, and are all second-order force contribution terms, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the tall structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (2) is the second-order component of the translational component, is the time derivative of the second-order component of the translational component, that is, the second-order component of the translational velocity, is the time derivative of the second-order component of the translational velocity, that is, the second-order component of the translational acceleration, {} is a 6×1 matrix;

[0018] The third-order equation of motion is:

[0019]

[0020] Where, is the third-order dynamic force in the wave force, and are all third-order force contribution terms, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the tall structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (3) is the third-order component of the translational component, is the time derivative of the third-order component of the translational component, that is, the third-order component of the translational velocity, is the time derivative of the third-order component of the translational velocity, that is, the third-order component of the translational acceleration, and {} is a 6×1 matrix.

[0021] Preferably, the wave force in S4 includes first-order force, second-order force and third-order force.

[0022] The first-order forces include and in:

[0023]

[0024] Where, is the first-order dynamic force, is the first-order restoring force, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, is the first-order component of the time derivative of the velocity potential, The first-order component of the displacement representing the heave motion of a tall structure, Represents the first-order component of the roll angle of a tall structure, Represents the first-order component of the pitch angle of a tall structure, (x0, y0, z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, It is a vertical upward vector logo;

[0025] The second-order forces include and in:

[0026]

[0027]

[0028] Where, is the second-order dynamic force, is the second-order restoring force, and are all second-order force contributions, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, and are the first-order and second-order components of the time derivative of the velocity potential, and are the first-order and second-order components of the heave displacement of the tall structure, and are the first-order and second-order components of the roll angle of the tall structure, and are the first-order component and the second-order component of the pitch angle of the tall structure, is the first-order component of the rotation angle of the towering structure, x0=(x0,y0,z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, x=(x,y,z) is the coordinate of the field point, is the Hamiltonian operator, is a vertically upward vector logo, C b is the cross section between the still water surface and the tall structure, dl is an infinitesimal length; φ (1) is the first-order component of the velocity potential, ξ (1)is the first-order component of the translation vector, α (1) is the first-order component of the rotation vector, η (1) is the first-order component of the wavefront rise;

[0029] The third-order forces include and in:

[0030]

[0031]

[0032] Where, is the third-order dynamic force, is the third-order restoring force, and are all third-order force contributions, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, and are the first-order, second-order and third-order components of the time derivative of the velocity potential, and are the first-order component, second-order component and third-order component of the heave displacement of the tall structure, and are the first-order component, second-order component and third-order component of the roll angle of the tall structure, and are the first-order component, second-order component and third-order component of the pitch angle of the tall structure, and are the first-order component and the second-order component of the rotation angle of the towering structure, x0=(x0,y0,z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, x=(x,y,z) is the coordinate of the field point in the calculated water area, is the Hamiltonian operator, is a vertically upward vector logo, C b is the cross section between the still water surface and the tall structure, dl is an infinitesimal length; φ (1) and φ (2) are the first-order and second-order components of the velocity potential, ξ (1) and ξ (2) are the first-order and second-order components of the translation vector, α (1) and α(2) are the first-order and second-order components of the rotation vector, η (1) and η (2) are the first-order and second-order components of the wavefront rise, Η (2) For correction items.

[0033] Preferably, the correction term H of the third-order force in S4 is (2) for

[0034]

[0035] Where, is the first-order component of the roll angle of the tall structure, is the first-order component of the pitch angle of a tall structure, It is the first-order component of the rotation angle of the tall structure.

[0036] Preferably, the S2 specifically includes the following steps:

[0037] S21. Perform a third-order perturbation expansion on the translational and rotational components of the tall structure, as well as the velocity potential and wave surface rise in the calculation water area. Obtain a Stokes third-order expansion based on the first-order, second-order, and third-order components of the translational and rotational components, velocity potential, and wave surface rise.

[0038] S22. Under the assumption of micro-waves, the boundary conditions in the calculation water area are expanded by Taylor series, thereby simplifying the calculation water area to include the bottom surface S D , still water surface S f The surface S of the object in equilibrium with the tall structure b Therefore, the boundary conditions of the calculation water area include bottom surface conditions, still water surface conditions, and surface conditions of tall structures in equilibrium positions.

[0039] S23. Based on the third-order expansion of Stokes obtained in S21 and the size information and wave information obtained in S1, the bottom surface conditions, still water surface conditions and surface conditions of the tall structure in equilibrium position are set for the calculation water area, so as to establish a nonlinear time-domain mathematical model of the interaction between waves and tall structures.

[0040] Preferably, the numerical integration method in S5 is a fourth-order Adams-Bashforth-Moultn prediction-correction method.

[0041] The technical solution adopted by the present invention to solve the above-mentioned second technical problem is: a computer-readable storage medium, characterized in that: the computer-readable storage medium stores a computer program for the high-frequency resonance response of waves and tall structures, and the computer program for the high-frequency resonance response of waves and tall structures can implement the above-mentioned calculation method when executed by a processor.

[0042] Compared with the prior art, the advantages of the present invention are: the present invention establishes a nonlinear time-domain mathematical model of the interaction between waves and tall structures based on the perturbation expansion theory, and the nonlinear time-domain mathematical model only needs to assemble a matrix once in the time domain, which can not only study the high-frequency resonance response of waves and tall structures, but also improve the computational efficiency of model tests; and the nonlinear time-domain mathematical model can also take into account various nonlinear influences, thereby making the motion equations of the high-frequency resonance response of waves and tall structures calculated by the nonlinear time-domain mathematical model more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 This is a flow chart of Example 1 of the present invention;

[0044] Figure 2 Schematic diagram of the model of the test example of the present invention;

[0045] Figure 3 1 is a time history diagram of the dimensionless pitch response of a single pile under four different incident wave amplitude conditions in the test example of the present invention;

[0046] Figure 4 for Figure 3 Amplitude spectrum of the time history curve. DETAILED DESCRIPTION

[0047] The present invention will be described in further detail below with reference to the accompanying drawings and embodiments.

[0048] Example 1

[0049] like Figure 1 As shown, this embodiment is a method for calculating the high-frequency resonance response of waves and tall structures.

[0050] In this embodiment, for the motion problem of a tall structure in waves, the tall structure is usually regarded as a rigid body with six degrees of freedom. The motion of the three-dimensional rigid body is described by motion components in six directions, among which ξ = (ξ1, ξ2.ξ3) are three translational components, representing the displacements of surge, sway and heave respectively; α = (α1, α2, α3) = (ξ4, ξ5.ξ6) are three rotation angles, representing the angles of roll, pitch and rotation respectively.

[0051] The method of this embodiment includes the following steps

[0052] S1. Obtaining the size information of tall structures and calculating the wave information in the water area;

[0053] S2. Perform a third-order perturbation expansion on the translational and rotational components of the tall structure, as well as the velocity potential and wave surface rise in the calculation water area. Based on the third-order perturbation expansion and the size and wave information obtained in S1, set the boundary conditions of the calculation water area to establish a nonlinear time-domain mathematical model for studying the interaction between waves and tall structures. The nonlinear time-domain mathematical model includes the Laplace equation, the Bernoulli equation for unsteady flow, and the motion equation derived from the conservation of momentum theorem.

[0054] Among them, S2 specifically includes the following steps:

[0055] S21. Perform a third-order perturbation expansion on the translational and rotational components of the tall structure, as well as the velocity potential and wave surface rise in the calculation water area. Based on the first-order, second-order, and third-order components of the translational and rotational components, the velocity potential, and wave surface rise, the Stokes third-order expansion is obtained as follows:

[0056] ξ=εξ (1) +ε 2 ξ (2) +ε 3 ξ (3) +…

[0057] α=εα (1) +ε 2 α (2) +ε 3 α (3) +...

[0058]

[0059] Where ε is the wave steepness, ξ (1) ,ξ (2) and ξ (3) are the first-order component, second-order component and third-order component of the translation vector, α (1) , α (2) and α (3) are the first-order, second-order and third-order components of the rotation vector, and are the first-order, second-order and third-order components of the incident potential, and are the first-order, second-order and third-order components of the scattering potential, and are the first-order component, second-order component and third-order component of the incident component of the wavefront rise, and They are the first-order component, second-order component and third-order component of the scattered component of the wavefront rise;

[0060] S22. Under the assumption of micro-waves, the boundary conditions in the calculation water area are expanded by Taylor series, thereby simplifying the calculation water area to include the bottom surface S D , still water surface S f The surface S of the object in equilibrium with the tall structure b Therefore, the boundary conditions of the calculation water area include bottom surface conditions, still water surface conditions, and surface conditions of tall structures in equilibrium positions.

[0061] S23, based on the third-order expansion of Stokes obtained in S21 and the size information and wave information obtained in S1, set the bottom surface conditions of the calculation water area, the still water surface conditions and the surface conditions of the towering structure in the equilibrium position, that is, the bottom surface conditions are the bottom surface S of the calculation water area. D , the velocity of the fluid is zero, and the static water surface condition is that on the static water surface S f The pressure value on the surface is equal to the atmospheric pressure, and it is assumed that water protons cannot leave the static water surface S f , thus obtaining the static water surface condition, and the object surface condition is the object surface S when the towering structure is in the equilibrium position b The upper normal velocity is equal to the adjacent fluid velocity, thus establishing a nonlinear time-domain mathematical model of the interaction between waves and tall structures;

[0062] S3. According to the boundary conditions set in S2, the velocity potential of the fluid motion velocity distribution and change is calculated using the Laplace equation in the nonlinear time domain mathematical model, where the velocity potential φ is calculated from the incident potential φ i and scattering potential φ s composition,

[0063] Incident potential φ i The first-order component, second-order component and third-order component of

[0064]

[0065] Where ω (ω = 2π / T, T is the wave period) is the wave frequency, k (k = 2π / L, L is the wavelength) is the wave number, θ is the wave incident angle, A is the incident wave amplitude, g is the gravitational acceleration, d is the water depth, h is the wave height, t is the time, and x = (x, y, z) is the coordinate of the field point;

[0066] Using Laplace's equation and Green's second theorem, we can get the scattering potential φ s The boundary integral equation of the k-th order component (k=1, 2, 3) is

[0067]

[0068] Where G(x,x0) is the Green function, is the unit normal vector of the object surface of the towering structure in the equilibrium position pointing to the fluid in the calculation water area, x0=(x0,y0,z0) is the coordinate of the source point, x=(x,y,z) is the coordinate of the field point, ds represents an infinitesimal surface element, S is the boundary including the towering structure and the still water surface, where

[0069]

[0070] Where d is the water depth, x0 = (x0, y0, z0) is the coordinate of the source point, and x = (x, y, z) is the coordinate of the field point;

[0071] Through the above scattering potential φ s The boundary integral equation is combined with the calculated φ i , thus obtaining the velocity potential φ;

[0072] S4. Based on the velocity potential determined in S3, the non-steady Bernoulli equation in the nonlinear time domain mathematical model is used to calculate the pressure distribution on the surface of the tall structure. The pressure distribution is divided according to different orders to obtain the wave force of the high-frequency resonance response between the wave and the tall structure. The wave force includes first-order force, second-order force and third-order force.

[0073] The first-order forces include and in:

[0074]

[0075] Where, is the first-order dynamic force, is the first-order restoring force, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, is the first-order component of the time derivative of the velocity potential, The first-order component of the displacement representing the heave motion of a tall structure, Represents the first-order component of the roll angle of a tall structure, Represents the first-order component of the pitch angle of a tall structure, (x0, y0, z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, It is a vertical upward vector logo;

[0076] The second-order forces include and in:

[0077]

[0078] Where, is the second-order dynamic force, is the second-order restoring force, and are all second-order force contributions, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, and are the first-order and second-order components of the time derivative of the velocity potential, and are the first-order and second-order components of the heave displacement of the tall structure, and are the first-order and second-order components of the roll angle of the tall structure, and are the first-order component and the second-order component of the pitch angle of the tall structure, is the first-order component of the rotation angle of the towering structure, x0=(x0,y0,z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, x=(x,y,z) is the coordinate of the field point, is the Hamiltonian operator, is a vertically upward vector logo, C b is the cross section between the still water surface and the tall structure, dl is an infinitesimal length; φ (1) is the first-order component of the velocity potential, ξ (1) is the first-order component of the translation vector, α (1) is the first-order component of the rotation vector, η (1) is the first-order component of the wavefront rise;

[0079] The third-order forces include and in:

[0080]

[0081] Where, is the third-order dynamic force, is the third-order restoring force, and are all third-order force contributions, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, and are the first-order, second-order and third-order components of the time derivative of the velocity potential, and are the first-order component, second-order component and third-order component of the heave displacement of the tall structure, and are the first-order component, second-order component and third-order component of the roll angle of the tall structure, and are the first-order component, second-order component and third-order component of the pitch angle of the tall structure, and are the first-order component and the second-order component of the rotation angle of the towering structure, x0=(x0,y0,z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, x=(x,y,z) is the coordinate of the field point in the calculated water area, is the Hamiltonian operator, is a vertically upward vector logo, C b is the cross section between the still water surface and the tall structure, dl is an infinitesimal length; φ (1) and φ (2) are the first-order and second-order components of the velocity potential, ξ (1) and ξ (2) are the first-order and second-order components of the translation vector, α (1) and α (2) are the first-order and second-order components of the rotation vector, η (1) and η (2) are the first-order and second-order components of the wavefront rise, Η (2) is the correction item;

[0082] Among them, the correction term H (2) for

[0083]

[0084] Where, is the first-order component of the roll angle of the tall structure, is the first-order component of the pitch angle of a tall structure, It is the first-order component of the rotation angle of the tall structure.

[0085] S5, nonlinear time domain mathematical model: The wave force obtained in S4 is input into the motion equation, and the motion equation is solved by numerical integration method. The numerical integration method is the fourth-order Adams-Bashforth-Moultn prediction-correction method. The motion equation includes the first-order motion equation, the second-order motion equation and the third-order motion equation. The first-order motion equation is:

[0086]

[0087] Where, is the first-order dynamic force in the wave force, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the towering structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (1) is the first-order component of the translational component, is the time derivative of the first-order component of the translational component, that is, the first-order component of the translational velocity, is the time derivative of the first-order component of the translational velocity, i.e., the first-order component of the translational acceleration, and {} is a 6×1 matrix;

[0088] The second-order equation of motion is:

[0089]

[0090] Where, is the second-order dynamic force in the wave force, and are all second-order force contribution terms, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the tall structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (2) is the second-order component of the translational component, is the time derivative of the second-order component of the translational component, that is, the second-order component of the translational velocity, is the time derivative of the second-order component of the translational velocity, that is, the second-order component of the translational acceleration, {} is a 6×1 matrix;

[0091] The third-order equation of motion is:

[0092]

[0093] Where, is the third-order dynamic force in the wave force, and are all third-order force contribution terms, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the tall structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (3)is the third-order component of the translational component, is the time derivative of the third-order component of the translational component, that is, the third-order component of the translational velocity, is the time derivative of the third-order component of the translational velocity, that is, the third-order component of the translational acceleration, {} is a 6×1 matrix;

[0094] Among them, the expression of the mass matrix [M] is

[0095]

[0096] Where M is the mass of the tall structure, (x c ,y c ,z c ) are the coordinates of the center of mass of the towering structure, (x0, y0, z0) are the coordinates of the source point, is the moment of inertia of the tall structure, calculated as follows:

[0097]

[0098] Where V b is the volume of the tall structure, ρ b is the density of the tall structure, x i represents the i-th order component of the field point, x 0i Represents the i-order component of the source point, x j represents the j-order component of the field point, x 0j represents the j-order component of the source point, i, j = 1, 2, 3, dv represents an infinitesimal volume;

[0099] The damping matrix [B] is obtained through physical model tests and is a known quantity;

[0100] The expression of the restoring force matrix [C] is

[0101]

[0102] Where ρ is the density of water in the calculated water area, A wp To calculate the surface area of ​​the water area, M is the mass of the tall structure, g is the acceleration due to gravity, (x c ,y c ,z c ) is the coordinate of the center of mass of the towering structure, (x0, y0, z0) is the coordinate of the source point, about the water surface A and The calculation formula is as follows:

[0103]

[0104] Where x i represents the i-th order component of the field point, x 0iRepresents the i-order component of the source point, x j represents the j-order component of the field point, x 0j represents the j-order component of the source point, i, j = 1, 2, 3, ds represents an infinitesimal surface element;

[0105] Regarding the volume V of the displaced water and The calculation formula is as follows:

[0106]

[0107] Where x i represents the i-th order component of the field point, x 0i Represents the i-order component of the source point, x j represents the j-order component of the field point, x 0j Represents the j-order component of the source point, i, j = 1, 2, 3, dv represents an infinitesimal volume.

[0108] Example 2

[0109] A computer-readable storage medium stores a computer program for high-frequency resonance response of waves and tall structures. The computer program for high-frequency resonance response of waves and tall structures can implement the calculation method of Example 1 when executed by a processor.

[0110] Test example

[0111] In this experiment, a simplified single pile foundation model is designed in the calculation water area, such as Figure 2 As shown in the figure, the monopile is represented as a tall floating structure. The bottom of the monopile is hinged to the bottom of the calculation water area. The monopile can rotate around the hinge point. A spring extending in the horizontal direction is provided on the top of the monopile. The monopile in the calculation water area is simplified to a single degree of freedom system, that is, only the motion mode in the pitch direction is considered.

[0112] In this test case, four different sets of dimensionless incident wave amplitudes (A / R = 0.4, 0.5, 0.6 and 0.7) are used to study the effect of wave nonlinearity on the high-frequency response of a single pile, where A is the incident wave amplitude, defined as half of the incident wave height H, and R is the radius of the single pile. Figure 3 As shown in Figure 2, the dimensionless pitch response time history of a single pile under four different incident wave amplitude conditions is shown in Figure 2. Figure 4 for Figure 3 The amplitude spectrum of the time history curve shows that there are second-order and third-order motions.

Claims

1. A method for calculating the high-frequency resonance response of waves and tall structures, characterized by: The method comprises the following steps: S1. Obtaining the size information of tall structures and calculating the wave information in the water area; S2. Perform a third-order perturbation expansion on the translational and rotational components of the tall structure, as well as the velocity potential and wave surface rise in the calculation water area. Based on the third-order perturbation expansion and the size and wave information obtained in S1, set the boundary conditions of the calculation water area to establish a nonlinear time-domain mathematical model for studying the interaction between waves and tall structures. The nonlinear time-domain mathematical model includes the Laplace equation, the Bernoulli equation for unsteady flow, and the motion equation derived from the conservation of momentum theorem. S3. According to the boundary conditions set in S2, the velocity potential of the fluid motion velocity distribution and change is calculated using the Laplace equation in the nonlinear time domain mathematical model; S4. Based on the velocity potential determined in S3, the non-steady Bernoulli equation in the nonlinear time-domain mathematical model is used to calculate the pressure distribution on the surface of the tall structure. The pressure distribution is divided according to different orders to obtain the wave force of the high-frequency resonance response between the wave and the tall structure. S5, the nonlinear time-domain mathematical model inputs the wave force obtained in S4 into the motion equation, uses the numerical integration method to solve the motion equation, and predicts the motion response of waves and tall structures in the calculated water area in real time; the motion equation includes the first-order motion equation, the second-order motion equation, and the third-order motion equation. The first-order motion equation is: Where, is the first-order dynamic force in the wave force, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the towering structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (1) is the first-order component of the translational component, is the time derivative of the first-order component of the translational component, that is, the first-order component of the translational velocity, is the time derivative of the first-order component of the translational velocity, i.e., the first-order component of the translational acceleration, and {} is a 6×1 matrix; The second-order equation of motion is: Where, is the second-order dynamic force in the wave force, and are all second-order force contribution terms, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the tall structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (2) is the second-order component of the translational component, is the time derivative of the second-order component of the translational component, that is, the second-order component of the translational velocity, is the time derivative of the second-order component of the translational velocity, that is, the second-order component of the translational acceleration, {} is a 6×1 matrix; The third-order equation of motion is: Where, is the third-order dynamic force in the wave force, and are all third-order force contribution terms, [M], [B] and [C] are all 6×6 matrices, [M] is the mass matrix of the tall structure, [B] is the damping matrix, [C] is the restoring force matrix, ξ (3) is the third-order component of the translational component, is the time derivative of the third-order component of the translational component, that is, the third-order component of the translational velocity, is the time derivative of the third-order component of the translational velocity, that is, the third-order component of the translational acceleration, and {} is a 6×1 matrix.

2. The method for calculating high-frequency resonance response of waves and tall structures according to claim 1, characterized in that: The wave force in S4 includes first-order force, second-order force and third-order force. The first-order forces include and in: Where, is the first-order dynamic force, is the first-order restoring force, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, is the first-order component of the time derivative of the velocity potential, The first-order component of the displacement representing the heave motion of a tall structure, Represents the first-order component of the roll angle of a tall structure, Represents the first-order component of the pitch angle of a tall structure, (x0, y0, z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, It is a vertical upward vector logo; The second-order forces include and in: Where, is the second-order dynamic force, is the second-order restoring force, and are all second-order force contributions, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, and are the first-order and second-order components of the time derivative of the velocity potential, and are the first-order and second-order components of the heave displacement of the tall structure, and are the first-order and second-order components of the roll angle of the tall structure, and are the first-order component and the second-order component of the pitch angle of the tall structure, is the first-order component of the rotation angle of the towering structure, x0=(x0,y0,z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, x=(x,y,z) is the coordinate of the field point, is the Hamiltonian operator, is a vertically upward vector logo, C b is the cross section between the still water surface and the tall structure, dl is an infinitesimal length; φ (1) is the first-order component of the velocity potential, ξ (1) is the first-order component of the translation vector, α (1) is the first-order component of the rotation vector, η (1) is the first-order component of the wavefront rise; The third-order forces include and in: Where, is the third-order dynamic force, is the third-order restoring force, and are all third-order force contributions, ρ is the density of water in the calculation area, S b is the surface of the object in the equilibrium position of the towering structure, n = (n1, n2, n3) is the unit normal vector of the object surface in the equilibrium position of the towering structure pointing to the fluid in the calculation water area, ds represents an infinitesimal surface element, g is the acceleration of gravity, A wp To calculate the area of ​​the water surface, and are the first-order, second-order and third-order components of the time derivative of the velocity potential, and are the first-order component, second-order component and third-order component of the heave displacement of the tall structure, and are the first-order component, second-order component and third-order component of the roll angle of the tall structure, and are the first-order component, second-order component and third-order component of the pitch angle of the tall structure, and are the first-order component and the second-order component of the rotation angle of the towering structure, x0=(x0,y0,z0) is the coordinate of the source point, (x f ,y f ,z f ) is the center coordinate of the calculated water area, x=(x,y,z) is the coordinate of the field point in the calculated water area, is the Hamiltonian operator, is a vertically upward vector logo, C b is the cross section between the still water surface and the tall structure, dl is an infinitesimal length; φ (1) and φ (2) are the first-order and second-order components of the velocity potential, ξ (1) and ξ (2) are the first-order and second-order components of the translation vector, α (1) and α (2) are the first-order and second-order components of the rotation vector, η (1) and η (2) are the first-order and second-order components of the wavefront rise, Η (2) For correction items.

3. The method for calculating high-frequency resonance response of waves and tall structures according to claim 2, characterized in that: The correction term Η of the third-order force in S4 (2) for Where, is the first-order component of the roll angle of the tall structure, is the first-order component of the pitch angle of a tall structure, It is the first-order component of the rotation angle of the tall structure.

4. The method for calculating high-frequency resonance response of waves and tall structures according to claim 1, characterized in that: The S2 specifically includes the following steps: S21. Perform a third-order perturbation expansion on the translational and rotational components of the tall structure, as well as the velocity potential and wave surface rise in the calculation water area. Obtain a Stokes third-order expansion based on the first-order, second-order, and third-order components of the translational and rotational components, velocity potential, and wave surface rise. S22. Under the assumption of micro-waves, the boundary conditions in the calculation water area are expanded by Taylor series, thereby simplifying the calculation water area to include the bottom surface S D , still water surface S f The surface S of the object in equilibrium with the tall structure b Therefore, the boundary conditions of the calculation water area include bottom surface conditions, still water surface conditions, and surface conditions of tall structures in equilibrium positions. S23. Based on the third-order expansion of Stokes obtained in S21 and the size information and wave information obtained in S1, the bottom surface conditions, still water surface conditions and surface conditions of the tall structure in equilibrium position are set for the calculation water area, so as to establish a nonlinear time-domain mathematical model of the interaction between waves and tall structures.

5. The method for calculating high-frequency resonance response between waves and tall structures according to claim 1, characterized in that: The numerical integration method in S5 is a fourth-order Adams-Bashforth-Moultn prediction-correction method.

6. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program for high-frequency resonance response of waves and tall structures, and the computer program for high-frequency resonance response of waves and tall structures can implement the method according to any one of claims 1 to 5 when executed by a processor.

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