PEM / EPIM-FFT calculation method and system for non-stationary random seismic response of jacket platform

Abstract

The PEM / EPIM-FFT calculation method and system for the non-stationary random seismic response of a jacket platform belong to the technical field of dynamic analysis and random vibration calculation of marine engineering structures. The present application combines the virtual excitation method, the explicit precise integration method and the fast Fourier transform, and pre-combines the constraint reaction force restoring relationship, the base response synthesis relationship and the modal response convolution kernel to form a target response direct output kernel, so as to directly obtain the evolution power spectral density and time-varying standard deviation of the target node displacement and the base response. The present application can uniformly output the evolution power spectral density and time-varying standard deviation of the target node displacement and the base response in the same PEM / EPIM-FFT calculation framework, and is suitable for the demand for simultaneously performing engineering checking on the local displacement response and the overall base response in the seismic analysis of a jacket platform.

Description

Technical Field

[0001] This invention belongs to the field of dynamic analysis and random vibration calculation technology of marine engineering structures, and relates to the PEM / EPIM-FFT calculation method and system for non-stationary random seismic response of jacket platforms. Background Technology

[0002] Jacket platforms are crucial infrastructure in offshore oil and gas development, typically characterized by complex spatial linkages, numerous degrees of freedom, and significant low-order overall bending mode control. Under seismic loading, the platform's superstructure, pile foundation area, and base shear and overturning moment can exhibit pronounced time-varying response characteristics. The amplitude and frequency components of actual seismic ground motions evolve over time, exhibiting non-stationary stochastic properties. While virtual excitation methods can transform stochastic response analysis into solving multiple deterministic complex-valued dynamic response problems, for jacket platforms with numerous frequency integration points and long analysis time histories, state recursion, modal superposition, and physical response recovery are still necessary at a large number of frequency points. Explicit time-domain methods can represent the relationship between excitation history and response as a causal convolution, but directly accumulating all historical terms point-by-point in the time domain leads to a rapid increase in computational complexity with the number of sampling points over time. Furthermore, engineering evaluation of jacket platforms not only focuses on nodal displacements but also on base shear, axial force, overturning moment, and torque derived from constraint reaction forces. Therefore, a computational method is needed that can combine non-stationary virtual excitation, explicit fine integration, FFT convolution acceleration, and direct output of base reaction force to efficiently obtain the evolution power spectral density (EPSD) and time-varying standard deviation of target nodal displacement and base response within the same analytical framework.

[0003] As offshore oil and gas development and offshore wind power projects expand into deeper and more complex sea areas, jacket platforms not only bear long-term environmental loads such as waves, currents, and wind loads during their service life, but may also be subjected to seismic ground motions with significant non-stationary characteristics. The intensity, frequency components, and duration of seismic ground motions vary over time. Jacket platforms also have characteristics such as complex spatial trusses, a large number of degrees of freedom, significant modal coupling, and the importance of foundation constraint reactions for engineering verification. Therefore, their non-stationary stochastic seismic response analysis usually requires the simultaneous acquisition of multiple response statistics, including nodal displacements, base shear forces, axial forces, overturning moments, and torques. In existing engineering calculations, if time-domain integration combined with a large number of samples or frequency-by-frequency and time-step recursion is used, long-term dynamic solutions are often required repeatedly. The computational load increases rapidly with the number of frequency points, time steps, and retained modes. If conventional virtual excitation methods or modal superposition methods are used, although random responses can be transformed into deterministic responses, for the calculation of long-term evolution power spectral density under non-stationary excitation, there are still problems such as repeated recursion within frequency cycles, large overhead of accumulating convolutional history terms, and the need to recover the complete physical displacement / velocity / acceleration of the base response before post-processing. These problems make it difficult to meet the requirements of high efficiency, unified output, and engineering verifiability in the seismic analysis of jacket platforms.

[0004] To address the aforementioned issues, Chinese invention patent CN107491585B discloses a structural topology optimization design method with random displacement response variance as the objective. This method uses a large mass method to convert multi-point acceleration into forces applied to the structure, and employs virtual excitation and modal superposition methods to solve for the random displacement response, which is then used for structural topology optimization design. However, this approach focuses on the random displacement response variance as the topology optimization objective and does not address the calculation of the evolutionary power spectral density, explicit fine integral recursive expansion, FFT convolution acceleration, or direct output of the basement response during non-stationary random earthquakes affecting the jacket platform. Chinese invention patent CN110083920B discloses an analysis method for the random response of a non-proportional damped structure under seismic loading. This method obtains the unit impulse response of the structure with respect to arbitrary excitation, establishes a direct convolution formula for calculating the response power spectrum based on random vibration theory, and uses POD decomposition and FFT to calculate the evolving frequency response matrix, thereby obtaining the evolving power spectrum matrix and variance matrix of the structural response. While this approach is closer to the calculation of random responses in non-stationary earthquakes, its technical approach focuses on the convolution of the unit impulse response and the calculation of the evolving frequency response matrix of non-proportional damped structures. It does not consider the structural modeling characteristics of the hydrodynamic added mass of the jacket platform, nor does it incorporate the response quantities of marine platform engineering, such as the recovery of constrained degrees of freedom reaction forces, the resultant force of the base, and the synthesis of the resultant moment, into a unified target response output kernel.

[0005] In summary, how to solve the problem of unified and efficient calculation of long-time-history, multi-frequency, nodal displacement and base response in the non-stationary random seismic response of jacket platform, so as to efficiently obtain the evolution power spectral density and time-varying standard deviation of target nodal displacement and base response within the same analysis framework, has become an urgent problem to be solved. Summary of the Invention

[0006] The technical problem this invention aims to solve is the large computational burden of frequency-wise and time-step-wise state recursion in the analysis of multi-frequency, long-time-history non-stationary random seismic responses of jacket platforms, and the difficulty in uniformly and efficiently outputting target node displacements and base responses within the same random vibration analysis framework. This invention provides a PEM / EPIM-FFT calculation method and system for non-stationary random seismic responses of jacket platforms. This invention combines the virtual excitation method, explicit fine integration method, and fast Fourier transform, and pre-combines the constraint reaction force recovery relationship, the base response synthesis relationship, and the modal response convolution kernel to form a direct target response output kernel, thereby directly obtaining the evolutionary power spectral density and time-varying standard deviation of the target node displacement and base response.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform includes the following steps: Step 1: Obtain the structure, environment, seismic input, discrete parameters, and target response parameters of the jacket platform; calculate the hydrodynamic added mass for the members below the water surface, and sequentially form the total structural mass matrix, Rayleigh damping matrix, and structural relative motion equations to obtain the basic dynamic model. Details are as follows: Step 1.1: Obtain the complete structural mass matrix, complete stiffness matrix, node coordinates, member connection relationships, member outer diameter, member length, immersion range, constrained degrees of freedom, seismic input direction, structural damping ratio, time discretization parameters, frequency discretization parameters, non-stationary ground motion parameters, target nodes and their displacement directions, support nodes, and base reference points of the finite element model of the jacket platform. In this invention, all structural degrees of freedom of the finite element model of the jacket platform are divided into unconstrained degrees of freedom and constrained degrees of freedom. The unconstrained degrees of freedom refer to the degrees of freedom for which no boundary constraints are applied and whose relative displacement response needs to be solved through the structural dynamic equilibrium equations; the constrained degrees of freedom refer to the degrees of freedom for which fixed constraints, hinged constraints, or other boundary constraints are applied, and whose relative displacement is determined by the corresponding boundary conditions. According to the degree of freedom division method, the structural dynamic response is solved on the unconstrained degrees of freedom, and the support reaction force at the constraint position is recovered according to the dynamic equilibrium equations corresponding to the constrained degrees of freedom.

[0008] Step 1.2: Based on the node coordinates, member connection relationships, member outer diameters, member lengths, and immersion intervals obtained in Step 1.1, the inertial force per unit length of the submerged member is calculated using the Morison equation's inertial force term, and the equivalent nodal hydrodynamic mass of each member element is obtained accordingly. e The hydrodynamic inertial force per unit length of a circular tube element is expressed as: (1) in, For the first e Each rod element in depth coordinates z and time t Dynamic inertial force per unit length at a given point; subscript It represents water; e Number the member units; The hydrodynamic inertia coefficient; The density of seawater; Pi; No. e The outer diameter of each bar element; For the first e Each rod element in depth coordinates z The relative acceleration at that point.

[0009] The first e When the hydrodynamic added mass of a single member element is evenly distributed to the two end nodes, the equivalent hydrodynamic added mass of a single end node is expressed as: (2) in, For the first e The hydrodynamic additional mass of each member element is allocated to a single-end node; For the first e The length of each bar element.

[0010] Step 1.3: Based on the equivalent nodal hydrodynamic mass of each member element, assemble them into a hydrodynamic mass matrix according to the nodal degrees of freedom. Superimpose this matrix with the structural mass matrix to obtain the total structural mass matrix used for unconstrained degree-of-freedom dynamic solutions. (3) in, The total mass matrix of the structure; This is the structural mass matrix corresponding to the unconstrained degrees of freedom; This represents the hydrodynamic added mass matrix corresponding to the unconstrained degrees of freedom; subscript This represents the structural body. During assembly, the complete total mass matrix of the structure, including the constrained degrees of freedom, is preserved for restoring the constraint reaction forces in step 5.

[0011] Step 1.4: Based on the complete stiffness matrix and structural damping ratio obtained in Step 1.1, and the total structural mass matrix obtained in Step 1.3, first solve the undamped characteristic equation to obtain the first and second natural circular frequencies used for damping calibration, and then construct the damping matrix using Rayleigh damping: (4) (5) (6) in, This is the damping matrix corresponding to the unconstrained degrees of freedom; This is the stiffness matrix corresponding to the unconstrained degrees of freedom; Rayleigh's mass proportional damping coefficient; This refers to the Rayleigh stiffness proportional damping coefficient; The structural damping ratio; The first natural circular frequency of the structure; This is the second-order natural circular frequency. When calculating the damping matrix, the same... and Construct a complete damping matrix that includes the constrained degrees of freedom.

[0012] Step 1.5: Based on the total structural mass matrix obtained in Step 1.3, the damping matrix and stiffness matrix obtained in Step 1.4, and the seismic input direction obtained in Step 1.1, establish the structural relative motion equations for the hydrodynamically added mass: (7) in, The relative displacement vector of the structural nodes; It is a relative velocity vector; It is a relative acceleration vector; Input the direction vector for the earthquake; Ground acceleration time history; subscript Indicates ground sports markings.

[0013] Steps 1.1 to 1.5 sequentially yield the total mass matrix, damping matrix, stiffness matrix, seismic input direction vector, and relative motion equations of the structure, thus forming the basic dynamic model used in subsequent virtual excitation construction, modal decoupling, and constraint reaction recovery.

[0014] Step 2: Based on the fundamental dynamic model obtained in Step 1, firstly, non-stationary virtual ground accelerations at each discrete frequency point are constructed. Then, the mass-normalized mode shapes are solved, and modal coordinate transformations are established. Finally, the virtual generalized coordinate equations are obtained through successive decoupling. Specifically, as follows: Step 2.1: Based on the non-stationary ground motion parameters obtained in Step 1.1, the ground acceleration is expressed as the product of a deterministic intensity envelope function and a zero-mean stationary Gaussian process, yielding the evolutionary power spectral density of the ground acceleration: (8) (9) in, The deterministic intensity envelope function; It is a zero-mean stationary Gaussian process; for The evolution of power spectral density; This is the angular frequency variable.

[0015] Step 2.2: Based on the evolved power spectral density and deterministic intensity envelope function obtained in Step 2.1, and based on the frequency discrete parameters obtained in Step 1.1, the virtual excitation method is used to construct the first... h Modulated harmonic virtual ground acceleration at discrete frequency points: (10) in, For the first h Virtual ground acceleration corresponding to each discrete frequency point; h Number the discrete frequency points. , This represents the total number of discrete frequency points. For the first h A discrete circular frequency; i is the imaginary unit.

[0016] Step 2.3, in parallel with steps 2.1 and 2.2, based on the total structural mass matrix and stiffness matrix obtained in step 1, solve the characteristic equation of undamped free vibration considering the hydrodynamic added mass, and obtain the mode shape matrix and the natural circular frequencies of each order: (11) (12) (13) in, For the first Mode shape vector; Modal order numbering, ; To preserve the modal order; For the first The natural circular frequency of the first order; To preserve the mode shape matrix of the modal composition; T Indicates matrix transpose; for An identity matrix of order 1.

[0017] Step 2.4, based on the mode shape matrix obtained in Step 2.3, the first... By projecting the virtual displacement response corresponding to each discrete frequency point onto modal coordinates, the relationship between the virtual physical displacement and the virtual generalized coordinates can be obtained: (14) in, For the first The virtual physical displacement response vector corresponding to each discrete frequency point; It is a virtual generalized coordinate vector composed of all preserved modes; For the first Virtual generalized coordinates.

[0018] Step 2.5: Based on the virtual ground acceleration obtained in Step 2.2 and the modal coordinate transformation obtained in Step 2.4, decouple the structural relative motion equation (7) in Step 1.5 to obtain the first... Virtual generalized coordinate equations for step-by-step decoupling: (15) in, For the first Virtual generalized velocity; For the first Order virtual generalized acceleration; For the first First-order modal damping ratio; For the first The mode participation factor is expressed as: (16) The non-stationary virtual ground acceleration at each discrete frequency point is obtained from steps 2.1 and 2.2. The modal parameters and modal coordinate transformation are obtained from steps 2.3 and 2.4. Then, step 2.5 transforms the multi-degree-of-freedom non-stationary random seismic response problem into a deterministic complex-valued single-degree-of-freedom response problem at different discrete frequency points and different modal orders.

[0019] Step 3: Based on the sequentially decoupled virtual generalized coordinate equations obtained in Step 2, establish the first-order modal state-space equations, and pre-calculate the modal state transition matrix and load integral vector using the explicit precise integration method to obtain the reusable state transition quantities outside the frequency cycle; specifically as follows: Step 3.1, based on the result obtained in step 2.5 j The virtual generalized coordinate equations define a modal state vector consisting of virtual generalized displacements and virtual generalized velocities: (17) And the first The first-order virtual generalized coordinate equations can be rewritten as first-order modal state-space equations: (18) in, For the first First-order modal state vector; The first-order time derivative of the modal state vector; For the first The first-order modal state matrix; Let the load input vectors be represented as follows: (19) (20) Step 3.2: Based on the time discretization parameters obtained in Step 1.1, discretize the time interval into... In time step The virtual ground acceleration is approximated using the zero-order hold-at-the-right endpoint, and the first-order modal state-space equations obtained in step 3.1 are explicitly and precisely integrated to obtain the single-step state transition relation: (twenty one) in, For the first A discrete time point; Number the time steps. ; To count the number of valid time points after removing the initial time, the index is... Indicates the effective length; For time step; For the first The modal state transition matrix of order ; For the first The load integral vectors of order are respectively expressed as: (twenty two) (twenty three) in, For the time variable in the load integration process.

[0020] because and It is only related to the structural modal parameters and time step, and not to the discrete frequency points. It is independent, so it can be pre-calculated and stored once before the frequency cycle begins, and reused at all discrete frequency points.

[0021] Step 4: Based on the modal state transition matrix and load integral vector obtained in Step 3, the modal state response is recursively expanded under zero initial conditions, yielding the modal displacement convolution kernel, modal velocity convolution kernel, and causal discrete convolution expression in sequence. Details are as follows: Step 4.1, under the conditions of zero initial displacement and zero initial velocity, recursively expand the single-step state transition relationship in Step 3.2 to obtain the... The causal discrete convolution expression for the first-order modal state response: (twenty four) in, For the first h A discrete circular frequency point; In the first h Discrete circular frequency points Under the corresponding virtual ground acceleration, the first j The first mode in the 1st order discrete time points The modal state vector; Numbering historical time points; For the first A historical point in time; The modal state transition matrix Power; For the first h Discrete circular frequency points The corresponding virtual ground acceleration is at the 1st Historical moments The complex value amplitude.

[0022] Step 4.2, based on the modal state response obtained in Step 4.1, define... The current time point and historical time points The discrete time lag positions between are determined, and the first and second components of the state vector are extracted using the displacement selection vector and velocity selection vector, respectively, to obtain the first... The first mode in the 1st order Modal displacement convolution kernel and modal velocity convolution kernel at each time lag: (25) (26) in, Numbering for time lag; For the first The first mode in the 1st order Modal displacement convolution kernel at a time lag; For the first The first mode in the 1st order Modal velocity convolution kernels at time lags; The modal state transition matrix Power; Choose a vector for the displacement; To select a vector for velocity, the two are represented as follows: (27) (28) Step 4.3: Based on the modal displacement convolution kernel and modal velocity convolution kernel obtained in Step 4.2, the first... The virtual generalized displacement and virtual generalized velocity are represented as a causal discrete convolution between the virtual ground acceleration history and the convolution kernel: (29) (30) in, In the first h Under the action of virtual ground acceleration corresponding to the discrete circular frequency, the first j The first mode at the current time point Virtual generalized displacement; For the first j The first mode in the 1st order r Modal displacement convolution kernel at each time lag position; In the first h Under the action of virtual ground acceleration corresponding to the discrete circular frequency, the first j The first mode at the current time point Virtual generalized velocity; For the first j The first mode in the 1st order r Modal velocity convolution kernels at time lag positions.

[0023] From step 4.1 to step 4.3, the time-step modal state recursion is transformed into an explicit causal discrete convolution between the virtual ground acceleration history and the modal convolution kernel, providing the basis for constructing the target response direct output kernel in step 5 and implementing FFT convolution in step 6.

[0024] Step 5: Based on the modal displacement convolution kernel and modal velocity convolution kernel obtained in Step 4, construct a direct output kernel for the target response, which is oriented towards the target node displacement and the basis response. Specifically: Step 5.1: Based on the modal displacement convolution kernels and modal velocity convolution kernels obtained in Step 4.2, combine them according to modal order to obtain the... Modal displacement convolution kernel vector and modal velocity convolution kernel vector at each time lag: (31) (32) in, For the first A column vector consisting of all preserved modal shift convolution kernels at a time lag; For the first A column vector consisting of all preserved modality velocity convolution kernels at each time lag; For the first-order mode in the... Modal displacement convolution kernel at a time lag; For the second-order mode in the 1st... Modal displacement convolution kernel at a time lag; For the first The first mode in the 1st order Modal displacement convolution kernel at a time lag; For the first-order mode in the... Modal velocity convolution kernels at time lags; For the second-order mode in the 1st... Modal displacement convolution kernel at a time lag; For the first The first mode in the 1st order Modal velocity convolution kernels at time lags.

[0025] Step 5.2: Based on the modal displacement convolution kernel vector obtained in Step 5.1, the target node mode shape output relationship is pre-combined using the target node displacement selection matrix to obtain the first... The target node displacement at the time lag is directly output as the kernel: (33) in, For the first The target node displacement at the time lag point is directly output as the kernel; A target node displacement selection matrix is ​​used to extract pre-specified target nodes and displacement directions from unconstrained physical displacements; subscripts This indicates the displacement output identifier.

[0026] Step 5.3: Divide the complete mass matrix, damping matrix, and stiffness matrix containing the constrained degrees of freedom into blocks according to the unconstrained and constrained degrees of freedom, to obtain the block matrix: (34) (35) in, The complete mass matrix containing the constrained degrees of freedom; This is the complete stiffness matrix including constrained degrees of freedom; the subscript f indicates the unconstrained degree of freedom number; the subscript c indicates the constrained degree of freedom number. This is the mass coupling matrix corresponding to the constrained degree-of-freedom rows and the unconstrained degree-of-freedom columns; This is the mass coupling matrix corresponding to the unconstrained degree-of-freedom rows and constrained degree-of-freedom columns; This is the mass coupling matrix corresponding to the unconstrained degree-of-freedom rows and columns; This is the mass coupling matrix corresponding to the constrained degree-of-freedom rows and columns; This is the stiffness coupling matrix corresponding to the constrained degree-of-freedom rows and the unconstrained degree-of-freedom columns; This is the stiffness coupling matrix corresponding to the unconstrained degree-of-freedom rows and constrained degree-of-freedom columns; This is the stiffness coupling matrix corresponding to the unconstrained degree-of-freedom rows and columns; This is the stiffness coupling matrix corresponding to the constrained degree-of-freedom rows and columns.

[0027] For the Rayleigh damped system obtained in step 1.4, the damping coupling matrix corresponding to the constrained degree-of-freedom row and the unconstrained degree-of-freedom column is: (36) in, This is the damping coupling matrix corresponding to the constrained degree-of-freedom rows and unconstrained degree-of-freedom columns in the complete damping matrix.

[0028] Step 5.4: Based on the block matrix obtained in Step 5.3, recover the virtual reaction force vector of the constrained degrees of freedom from the dynamic equilibrium equations corresponding to the constrained degrees of freedom: (37) in, This refers to the virtual reaction force vector at the constrained degrees of freedom; A virtual generalized acceleration vector that preserves all modes; The virtual generalized velocity vector that preserves all modes; A virtual generalized displacement vector that preserves all modes; Let be the direct inertial influence vector of ground acceleration on constraint reaction force, expressed as: (38) in, This represents the seismic input direction vector corresponding to the unconstrained degrees of freedom. The seismic input direction vector is the one corresponding to the constrained degrees of freedom.

[0029] Step 5.5: Based on the virtual reaction force vectors at the constrained degrees of freedom obtained in Step 5.4, extract the virtual reaction forces and virtual reaction moments of each support node according to the order of the support node degrees of freedom, and synthesize them into the resultant force and resultant moment of the base with respect to the base reference point. The resultant force and resultant moment of the base are then expressed as: (39) (40) in, The resultant force vector of the basis; For the first Virtual reaction force vectors of each support node; Number the support nodes; This represents the total number of support nodes; The resultant moment vector of the basis forces; For the first The virtual reaction moment vector of each support node; Represents the cross product of vectors; For the first The position vector of each support node relative to the base reference point; subscript This indicates the basis response identifier.

[0030] Step 5.6: Based on the resultant force and moment of the base obtained in Step 5.5, construct the base response vector according to a fixed order of resultant force in the three directions first, followed by resultant moment in the three directions: (41) in, The basis virtual response vector; , and These are the resultant base forces in the X, Y, and Z directions, respectively; , and These are the resultant moments of the base forces about the X, Y, and Z directions, respectively. The basis response synthesis matrix is ​​used to map the virtual reaction force vectors of constrained degrees of freedom to the basis response vectors.

[0031] Step 5.7: Based on the constrained degrees of freedom virtual reaction force vector and basis response vector obtained in steps 5.4 to 5.6, define the transformation matrices of modal acceleration, modal velocity, modal displacement, and ground acceleration direct terms to the basis response, respectively: (42) (43) (44) (45) in, This is the transformation matrix from modal acceleration to basis response; This is the transformation matrix from modal velocity to basis response; This is the transformation matrix from modal displacement to basis response; This is the transformation vector from the direct term of ground acceleration to the base response.

[0032] Step 5.8, based on the transformation matrix obtained in step 5.7 , , , The virtual generalized acceleration vector is eliminated using the modal dynamic equations obtained in step 2.5, so that the basis response is determined only by the virtual generalized displacement. Virtual generalized speed and virtual ground acceleration Define the modal damping diagonal matrix and the modal frequency squared diagonal matrix: (46) (47) in, This is the diagonal matrix of modal damping; This indicates that a diagonal matrix is ​​constructed from the elements within the parentheses; It is a diagonal matrix of modal frequencies squared; The damping ratio for the first-order mode; For the first First-order modal damping ratio; This is the first-order natural circular frequency; For the first The natural circular frequency.

[0033] The virtual generalized acceleration vector, composed of the retained modal dynamic equations, is represented as follows: (48) in, This is the vector of modal participation coefficients. This is the participation factor for the first-order mode shape. This is the participation factor for the second-order mode. For the first Mode participation factor.

[0034] Substituting equation (48) into equation (41), we obtain the basis response that does not explicitly include virtual generalized acceleration: (49) in, This is the basis response transformation matrix corresponding to the modal displacement term; This is the basis response transformation matrix corresponding to the modal velocity term; The basis response transformation vectors corresponding to the direct terms of virtual ground acceleration are expressed as follows: (50) (51) (52) Step 5.9, based on the modal displacement convolution kernel vector obtained in Step 5.1 and modal velocity convolution kernel vector and the basis response transformation matrix obtained in step 5.8 , , Construct the first The basis response at each time lag directly outputs the kernel: (53) in, For the first The basis response at a time lag directly outputs the kernel; For the Kronecker function, when hour ,when hour .

[0035] Step 5.10: Directly output the kernel based on the target node displacement obtained in Step 5.2. The basis response obtained in step 5.9 is directly output as the kernel. By vertically combining the two parallel output branches in a fixed order, we obtain the first... r The target response direct output kernels at each time lag are formed into a sequence of target response direct output kernels in order of time lag: (54) (55) in, For the first The target response at the time lag point is directly output as the kernel; The kernel sequence is directly output to arrange the target responses in order of time lag. The kernel is directly output for the target response at the first time lag. The kernel is directly output as the target response at the second time lag. For the first The target response at a time lag is directly output as the kernel; subscript Indicates the target response identifier.

[0036] Therefore, the target node displacement and the basis response can both be obtained directly from the same target response output kernel.

[0037] Step 6: Directly output the kernel based on the target response obtained in Step 5. The virtual ground acceleration at each discrete frequency point obtained in step 2 Under the condition of satisfying the linear convolution length requirement, fast Fourier transform, element-wise multiplication in the frequency domain, and inverse fast Fourier transform are performed to obtain the virtual response of the target node displacement and the virtual response of the basis within the effective time period; as follows: Step 6.1, based on the number of valid time points obtained in Step 4 Determine the Fast Fourier Transform length that is not less than the length of the complete linear convolution: (56) in, The length of the Fast Fourier Transform; For two lengths The length of the sequence after performing a full linear convolution; Indicates rounding up to the nearest integer; This indicates the Fast Fourier Transform identifier.

[0038] Step 6.2, based on the virtual ground acceleration obtained in Step 2.2 Construct the first according to the effective time point sequence The effective time history vector of virtual ground acceleration corresponding to each discrete frequency point: (57) in, For the first The effective time history vector of virtual ground acceleration corresponding to each discrete frequency point; For the first The complex amplitude of the virtual ground acceleration corresponding to each discrete frequency point at the first effective discrete time point; For the first The complex amplitude of the virtual ground acceleration corresponding to each discrete frequency point at the second effective discrete time point; For the first The virtual ground acceleration corresponding to the discrete frequency point, at the th discrete frequency point, is... Complex amplitude at each effective discrete time point.

[0039] Step 6.3: Directly output the kernel sequence based on the target response obtained in step 5.10. And the virtual ground acceleration effective time history vector obtained in step 6.2 Zeros are padded at the ends of both time series to extend their time dimensions to the length of the Fast Fourier Transform determined in step 6.1. Then, the Fast Fourier Transform, element-wise multiplication in the frequency domain, and Inverse Fast Fourier Transform are performed sequentially along the time dimension. The required effective time segment is extracted from the Inverse Fast Fourier Transform result to obtain the first... The virtual time history matrix of the target response corresponding to each discrete circular frequency point: (58) in, For the first The target response virtual time history matrix corresponds to each discrete circular frequency point, with each row corresponding to a target response component and each column corresponding to an effective time point. Indicates length is Fast Fourier Transform; Indicates length is The inverse fast Fourier transform; This indicates element-wise multiplication in the frequency domain; Indicates the truncation before One effective time point.

[0040] From steps 6.1 to 6.3, the target response is directly output as the kernel. With virtual ground acceleration effective time history vector The time-domain linear convolution between them is converted into frequency-domain multiplication, thereby directly obtaining the virtual time history of the target node displacement and the basis response for each discrete frequency point.

[0041] Step 7: Based on the target response virtual time history matrix obtained in Step 6, calculate the target response evolution power spectral density matrix and time-varying variance matrix sequentially, and complete the statistical output. Details are as follows: Step 7.1, based on the target response virtual time history matrix obtained in step 6.3 Calculate the first h The power spectral density matrix of the target response evolution corresponding to each discrete circular frequency point: (59) in, For the first h The time history matrix of the target response self-evolutionary power spectral density corresponding to each discrete circular frequency point; superscript Indicates complex conjugation.

[0042] Equation (59) gives the time history matrix of the self-evolving power spectral density of the target response in the th... h The values ​​at discrete circular frequency points. Let Represents the change in frequency with continuous circular frequency The changing target response self-evolves the power spectral density time history matrix, then For the matrix function in the th h Discrete circular frequency points Discrete values ​​at a given point.

[0043] Step 7.2: Based on the target response self-evolutionary power spectral density time history matrix obtained in Step 7.1, integrate along discrete frequency points to obtain the target response time-varying variance matrix: (60) in, The time-varying variance matrix of the target response; For continuous circular frequency variable The changing target response self-evolves power spectral density matrix; The starting angular frequency of the frequency discrete interval; It is the terminating angular frequency of the frequency discrete interval.

[0044] Step 7.3: Based on the target response time-varying variance vector obtained in step 7.2, take the square root of each element to obtain the time-varying standard deviation of the target node displacement and the basis response.

[0045] This invention also provides a PEM / EPIM-FFT calculation system for the non-stationary stochastic seismic response of a jacket platform. The system implements the aforementioned PEM / EPIM-FFT calculation method and includes a model input and dynamic modeling module, a non-stationary virtual excitation construction module, a modal decomposition and decoupling module, an EPIM pre-calculation module, a modal convolution kernel generation module, a target response direct output kernel construction module, an FFT response calculation module, and a statistical output module. Specifically: The model input and dynamic modeling module is used to obtain the structural mass matrix, stiffness matrix, nodal coordinates, constraint degrees of freedom, structural damping ratio, seismic input direction, time discretization parameters, frequency discretization parameters, target response parameters, and non-stationary ground motion parameters of the finite element model of the jacket platform. It then calculates the hydrodynamic added mass, the total mass matrix of the assembled structure, constructs the Rayleigh damping matrix, and establishes the structural relative motion equations. The dynamic basic model is then transferred to the modal decomposition and decoupling module and the target response direct output kernel construction module, and the non-stationary ground motion parameters are transferred to the non-stationary virtual excitation construction module.

[0046] The non-stationary virtual excitation construction module is used to construct a deterministic intensity envelope function and a stationary base process power spectral density based on the non-stationary ground motion parameters, and to construct a modulated harmonic virtual ground acceleration for each discrete frequency point, and to transmit the virtual ground acceleration to the mode decomposition and decoupling module and the FFT response calculation module.

[0047] The modal decomposition and decoupling module is used to solve the mass-normalized mode shape matrix and the natural circular frequencies of each order based on the total mass matrix and stiffness matrix of the structure, and to combine the virtual ground acceleration with the modal coordinate transformation to generate virtual generalized coordinate equations that are decoupled step by step; the modal parameters and mode participation coefficients are passed to the EPIM pre-calculation module, the modal convolution kernel generation module and the target response direct output kernel construction module.

[0048] The EPIM pre-computation module is used to calculate and store the modal state transition matrix and load integral vector of each order based on the virtual generalized coordinate equation and time step of successive decoupling; and to pass the modal state transition matrix and load integral vector to the modal convolution kernel generation module.

[0049] The modal convolution kernel generation module is used to generate modal displacement convolution kernels and modal velocity convolution kernels in a time lag order based on the modal state transition matrix, load integral vector, and mode participation coefficient; and to pass the modal convolution kernels to the target response direct output kernel construction module.

[0050] The target response direct output kernel construction module is used to construct the target node displacement direct output kernel and the basis response direct output kernel, and merge the two into a target response direct output kernel sequence according to a fixed response order; the target response direct output kernel sequence is then passed to the FFT response calculation module.

[0051] The FFT response calculation module is used to perform zero-padding, fast Fourier transform, frequency domain element-wise multiplication, inverse fast Fourier transform, and effective time period truncation on the target response kernel sequence and the virtual ground acceleration effective time history vector for each discrete frequency point to obtain the target response virtual time history matrix; and to pass the target response virtual time history matrix to the statistical output module.

[0052] The statistical output module is used to calculate the power spectral density matrix of the target response evolution based on the complex conjugate outer product of the virtual vector of the target response at each discrete frequency point, perform numerical integration along the circular frequency direction to obtain the time-varying variance vector of the target response, and output the time-varying standard deviation vector of the target node displacement and the basis response by taking the square root of each element.

[0053] The present invention also provides an electronic device, the electronic device including a processor and a memory, the memory storing a computer program, the computer program being executed by the processor to implement the above-described PEM / EPIM-FFT calculation method.

[0054] The present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described PEM / EPIM-FFT calculation method.

[0055] The beneficial effects of this invention are as follows: (1) This invention unifies non-stationary virtual excitation, explicit fine integration and fast Fourier transform into the same time-frequency hybrid analysis framework, transforms the non-stationary random seismic response problem of the jacket platform into a series of deterministic complex-valued response calculation problems, and converts long-term history convolution into frequency domain multiplication through FFT, effectively reducing the computational overhead in multi-frequency and long-term response analysis. (2) At the same time, the present invention moves the modal state transition matrix, load integral vector, modal displacement convolution kernel and modal velocity convolution kernel to the outside of the frequency cycle for pre-calculation, so that the virtual ground acceleration construction, fast Fourier transform, frequency domain corresponding element multiplication and inverse fast Fourier transform are mainly performed in the frequency cycle, thereby improving the computational efficiency under the condition of repeated frequency integral calculation. (3) Furthermore, the present invention incorporates the constraint degree of freedom equilibrium equation, support reaction force recovery relationship and base response synthesis relationship into the target response direct output kernel, avoiding the need to explicitly recover the complete physical displacement, velocity and acceleration time history at each discrete frequency point and then perform base response post-processing. Therefore, this invention can uniformly output the evolution power spectral density and time-varying standard deviation of target node displacement and base response within the same PEM / EPIM-FFT calculation framework, which is suitable for the need to simultaneously perform engineering verification of local displacement response and overall base response in seismic analysis of jacket platforms. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform in an embodiment of the present invention.

[0057] Figure 2 This is the finite element model of the four-legged guide frame platform in this embodiment of the invention.

[0058] Figure 3 This is a diagram of the natural frequency of the guide frame platform in an embodiment of the present invention.

[0059] Figure 4 This is the evolution power spectral density diagram of the X-direction displacement response of the first node in this embodiment of the invention.

[0060] Figure 5 This is the power spectral density diagram of the evolution of the X-direction displacement response of the second node in this embodiment of the invention.

[0061] Figure 6 This is the power spectral density diagram of the evolution of the X-direction displacement response of the third node in this embodiment of the invention.

[0062] Figure 7 This is the evolution power spectral density diagram of the X-direction displacement response of the fourth node in this embodiment of the invention.

[0063] Figure 8 This is the power spectral density diagram of the evolution of the X-direction base shear force in an embodiment of the present invention.

[0064] Figure 9 This is the power spectral density diagram of the evolution of the overturning moment around the Y-axis in an embodiment of the present invention.

[0065] Figure 10This is a comparison chart of the time-varying standard deviations of the node X-direction displacement calculated by three methods in this embodiment of the invention.

[0066] Figure 11 This is a comparison chart of the time-varying standard deviation of the X-direction base shear force calculated by three methods in the embodiments of the present invention.

[0067] Figure 12 This is a comparison chart of the time-varying standard deviations of the overturning moment around the Y-axis calculated by three methods in this embodiment of the invention.

[0068] Figure 13 This is a comparison chart of the calculation time of the three methods in the embodiments of the present invention.

[0069] Figure 14 This is a schematic diagram of the PEM / EPIM-FFT computing system structure in an embodiment of the present invention.

[0070] In the diagram: 1 First node, 2 Second node, 3 Third node, 4 Fourth node. Detailed Implementation

[0071] The present invention will be further described below with reference to specific embodiments.

[0072] In the following embodiments, the overall flow of the PEM / EPIM-FFT calculation method is as follows: Figure 1 As shown. Includes the following steps: Step 1: Obtain the structure, environment, seismic input, discrete parameters, and target response parameters of the jacket platform; calculate the hydrodynamic added mass for the members below the water surface, and sequentially form the total structural mass matrix, Rayleigh damping matrix, and structural relative motion equations to obtain the basic dynamic model. Details are as follows: Step 1.1: Obtain the complete structural mass matrix, complete stiffness matrix, node coordinates, member connection relationships, member outer diameter, member length, immersion range, constrained degrees of freedom, seismic input direction, structural damping ratio, time discretization parameters, frequency discretization parameters, non-stationary ground motion parameters, target nodes and their displacement directions, support nodes, and base reference points of the finite element model of the jacket platform. In this invention, all structural degrees of freedom of the finite element model of the jacket platform are divided into unconstrained degrees of freedom and constrained degrees of freedom. Unconstrained degrees of freedom refer to degrees of freedom without boundary constraints, whose relative displacement response needs to be solved through the structural dynamic equilibrium equations; constrained degrees of freedom refer to degrees of freedom with fixed constraints, hinged constraints, or other boundary constraints, whose relative displacement is determined by the corresponding boundary conditions. According to the degree of freedom division method, the structural dynamic response is solved on the unconstrained degrees of freedom, and the support reactions at the constraint locations are recovered according to the dynamic equilibrium equations corresponding to the constrained degrees of freedom.

[0073] The example uses a four-legged jacket platform as an example to perform a non-stationary random seismic response analysis. The finite element model of the four-legged jacket platform is as follows: Figure 2 As shown in the figure, 1, 2, 3, and 4 represent the first node 1, the second node 2, the third node 3, and the fourth node 4, respectively. The total structural height is 102.5m, the operating water depth is 79.5m, and there are two decks. The superstructure consists of columns and two decks. The columns are steel pipe components, and the two decks are H-shaped steel components. The upper deck is 20m long and 15m wide, and the lower deck is 18m long and 15m wide. The main body of the jacket structure is composed of main legs, horizontal struts, and staggered diagonal braces. The platform has four well slots, each with a water-tight sleeve. In the finite element modeling, spatial beam elements are used to discretize the jacket structure members. The structural mass matrix is ​​established using the lumped mass method, and the pile-soil interaction below the mud surface is simplified to a six-degree-of-freedom fully constrained bottom node.

[0074] In this embodiment, the finite element model of the jacket platform has a total of 1473 nodes, 821 elements, 3848 effective degrees of freedom, a structural dry mass of 1201.387t, a hydrodynamic added mass of 962.873t, and the coordinates of the structural centroid including the added water mass are (1.239, -0.013, -35.886)m. The above parameters are only used to illustrate the implementation process of this embodiment and do not constitute a limitation on the applicable objects and model scale of this invention. In this embodiment, the structural mass matrix and stiffness matrix are derived from the finite element model of the jacket platform, the constrained degrees of freedom are the six degrees of freedom constraints of the bottom support nodes of the platform, and the seismic input direction is taken as the global X direction of the platform. The time discretization parameters include an earthquake duration of 80s and a time step of 0.01s, and the frequency discretization parameters include a frequency integral range of 0–15 rad / s and a frequency step of 0.01 rad / s.

[0075] Step 1.2: Based on the node coordinates, member connection relationships, member outer diameters, member lengths, and immersion intervals obtained in Step 1.1, the inertial force per unit length of the submerged member is calculated using the Morison equation's inertial force term, and the equivalent nodal hydrodynamic mass of each member element is obtained accordingly. e The hydrodynamic inertial force per unit length of each circular tube element is calculated using formula (1). In this embodiment, the density of seawater is taken as 1025 kg / m³. 3 The inertial force coefficient is taken as 1.2. Since the diameter of the jacket members is relatively small compared to the characteristic scale of water disturbance under seismic action, this embodiment ignores the drag force term and only describes the influence of water on the dynamic characteristics of the structure in the form of the additional mass corresponding to the hydrodynamic inertial force.

[0076] The first eWhen the hydrodynamic additional mass of a single member element is evenly distributed to the two end nodes, the equivalent hydrodynamic additional mass of a single end node is calculated according to formula (2). When using the lumped mass method as an approximation, the first... e The total additional mass of each link element is evenly distributed to the two end nodes and assembled into the overall mass matrix according to the direction of the node degrees of freedom.

[0077] Step 1.3: Based on the equivalent node-added mass of the rod obtained in Step 1.2, assemble it into a hydrodynamic added mass matrix according to the direction of the node degree of freedom, and superimpose it with the structural mass matrix to obtain the total structural mass matrix for unconstrained degree of freedom dynamic solution according to formula (3).

[0078] Step 1.4: Based on the stiffness matrix and structural damping ratio obtained in Step 1.1 and the total structural mass matrix obtained in Step 1.3, first solve the undamped characteristic equation to obtain the first and second natural circular frequencies for damping calibration, and then construct the damping matrix using Rayleigh damping according to formulas (4) to (5).

[0079] In this embodiment, the structural damping ratio is set to 0.05. The Rayleigh damping coefficient is determined by the first two natural circular frequencies of the structure after considering the added mass. This process ensures that the damping matrix is ​​consistent with the total mass matrix and stiffness matrix of the structure, guaranteeing consistency in subsequent modal decomposition, modal damping calculation, and the establishment of the virtual generalized coordinate equation.

[0080] Step 1.5: Based on the total mass matrix of the structure obtained in Step 1.3, the damping matrix and stiffness matrix obtained in Step 1.4, and the seismic input direction obtained in Step 1.1, establish the relative motion equation of the structure with hydrodynamic added mass according to formula (7).

[0081] Steps 1.1 to 1.5 sequentially yield the total mass matrix, damping matrix, stiffness matrix, seismic input direction vector, and relative motion equations of the structure, thus forming the basic dynamic model used in subsequent virtual excitation construction, modal decoupling, and constraint reaction recovery.

[0082] Step 2: Based on the fundamental dynamic model obtained in Step 1, firstly, non-stationary virtual ground accelerations at each discrete frequency point are constructed. Then, the mass-normalized mode shapes are solved and modal coordinate transformations are established. Finally, the virtual generalized coordinate equations are obtained through successive decoupling. Specifically, as follows: Step 2.1: Based on the non-stationary ground motion parameters obtained in Step 1.1, the ground acceleration is expressed as the product of a deterministic intensity envelope function and a zero-mean stationary Gaussian process. The evolution power spectral density of the ground acceleration is obtained according to formulas (8) and (9).

[0083] In this embodiment, the non-stationary ground motion is described by a uniformly modulated evolution random process, and the intensity envelope function is a three-segment envelope function consisting of an ascent segment, a strong ground motion segment, and an attenuation segment, as shown in formula (61): (61) Among them, the end time of the rising segment Take 10 seconds as the end time of the stable phase in a strong earthquake. Take 30 seconds, attenuation coefficient Take 0.16.

[0084] The power spectral density of the stationary base process is obtained using the Kanai-Tajimi spectrum, as shown in equation (62): (62) Among them, the site soil excellent circular frequency Take 15.708 rad / s, site soil damping ratio Set to 0.6, spectral intensity factor Pick .

[0085] The above parameters are used to describe the time evolution and frequency distribution characteristics of the ground motion intensity in this embodiment.

[0086] Step 2.2: Based on the power spectral density and deterministic intensity envelope function obtained in Step 2.1, and based on the frequency discrete parameters obtained in Step 1.1, the virtual excitation method of formula (10) is used to construct the first... h Modulated simple harmonic virtual ground acceleration at discrete frequency points.

[0087] In this embodiment, the discrete frequency points are located in the range of 0–15 rad / s, with a frequency step size of 0.01 rad / s. For each discrete frequency point, a corresponding modulated harmonic virtual ground acceleration is constructed. Through this process, the original non-stationary random seismic response problem is transformed into a time-domain response calculation problem under a series of deterministic complex-valued virtual excitations, thereby avoiding repeated time history analysis of a large number of random samples.

[0088] In step 2.3, in parallel with steps 2.1 and 2.2, based on the total mass matrix and stiffness matrix of the structure obtained in step 1, solve the characteristic equation of undamped free vibration after considering the hydrodynamic added mass according to the format (11)~(13) to obtain the mode shape matrix and the natural circular frequencies of each order.

[0089] In this embodiment, Figure 3 The first 65 natural frequencies of the jacket platform are given; the number of modes is retained in this embodiment. Take the 65th order.

[0090] Step 2.4, based on the mode shape matrix obtained in Step 2.3, the first... The virtual displacement response corresponding to each discrete frequency point is projected onto the modal coordinates, and the relationship between the virtual physical displacement and the virtual generalized coordinates is obtained using formula (14).

[0091] In this embodiment, the first 65 mass-normalized mode shapes are used for modal superposition, and the target nodal displacement is extracted from the mode shape matrix through the target nodal displacement selection matrix.

[0092] Step 2.5: Based on the virtual ground acceleration obtained in Step 2.2 and the modal coordinate transformation obtained in Step 2.4, decouple the structural relative motion equation (7) in Step 1.5 to obtain the first... The virtual generalized coordinate equations of the order-by-order decoupling are shown in Equation (15).

[0093] The non-stationary virtual ground acceleration at each discrete frequency point is obtained from steps 2.1 and 2.2. The modal parameters and modal coordinate transformation are obtained from steps 2.3 and 2.4. Then, step 2.5 transforms the multi-degree-of-freedom non-stationary random seismic response problem into a deterministic complex-valued single-degree-of-freedom response problem at different discrete frequency points and different modal orders.

[0094] Step 3: Based on the sequentially decoupled virtual generalized coordinate equations obtained in Step 2, establish the first-order modal state-space equations, and pre-calculate the modal state transition matrix and load integral vector using the explicit precise integration method to obtain the reusable state transition quantities outside the frequency cycle; specifically as follows: Step 3.1, based on the result obtained in step 2.5 j The virtual generalized coordinate equation is defined as follows: the modal state vector is composed of virtual generalized displacement and virtual generalized velocity, as shown in formula (17).

[0095] In this embodiment, modal state-space equations are established for each of the 65 retained modes, as shown in Equation (18). Since each mode is a single-degree-of-freedom complex-valued response problem, its state vector consists only of virtual generalized displacement and virtual generalized velocity. Therefore, the dimensions of the subsequent state transition matrix and load integral vector are low, which facilitates unified pre-calculation and storage outside the frequency cycle.

[0096] Step 3.2: Based on the time discretization parameters obtained in Step 1.1, discretize the time interval into... In time step The virtual ground acceleration is approximated by the zero-order right endpoint, and the first-order modal state space equation obtained in step 3.1 is explicitly refined to obtain the single-step state transition relationship, as shown in formula (21).

[0097] In this embodiment, the earthquake duration is 80 seconds, the time step is 0.01 seconds, and the number of effective time points after removing the initial time is 8000. For each mode, the modal state transition matrix and load integral vector are only related to the natural circular frequency, modal damping ratio and time step of that mode, and are independent of the discrete frequency point. The modal state transition matrix and load integral vector corresponding to the 65th mode can be pre-calculated once before the frequency cycle begins, as shown in formulas (22) and (23).

[0098] Since the modal state transition matrix and load integral vector are only related to the structural modal parameters and time step, and are independent of the discrete frequency points... It is independent of frequency, so it can be pre-calculated and stored once before the frequency cycle begins, and can be reused at all discrete frequency points.

[0099] Step 4: Based on the modal state transition matrix and load integral vector obtained in Step 3, the modal state response is recursively expanded under zero initial conditions, yielding the modal displacement convolution kernel, modal velocity convolution kernel, and causal discrete convolution expression in sequence. Details are as follows: Step 4.1, under the conditions of zero initial displacement and zero initial velocity, recursively expand the single-step state transition relationship in Step 3.2 to obtain the... The causal discrete convolution expression for the first-order modal state response is shown in Equation (24).

[0100] In this embodiment, the initial relative displacement and initial relative velocity of the structure are both set to zero. By expanding the state transition relationship, the modal state response at any time point can be represented as a linear superposition of the virtual ground acceleration contributions at previous time points. This linear superposition is causal, meaning that the response at the current moment is determined only by the virtual excitations at the current and historical moments.

[0101] Step 4.2, based on the modal state response obtained in Step 4.1, define... The current time point and historical time points The discrete time lag positions between are determined, and the first and second components of the state vector are extracted using the displacement selection vector and velocity selection vector, respectively, to obtain the first... The first mode in the 1st order The modal displacement convolution kernel and modal velocity convolution kernel at each time lag are shown in equations (25) and (26).

[0102] In this embodiment, the modal participation coefficients are pre-incorporated into the modal displacement convolution kernel and the modal velocity convolution kernel. For each mode, a modal displacement convolution kernel and a modal velocity convolution kernel of length 8000 are generated. Since the modal state transition matrix, load integral vector, and modal participation coefficients are all independent of the virtual excitation frequency, the modal convolution kernels of each order can also be generated and stored once before the frequency cycle.

[0103] Step 4.3: Based on the modal displacement convolution kernel and modal velocity convolution kernel obtained in Step 4.2, the first... The virtual generalized displacement and virtual generalized velocity are represented as causal discrete convolutions between the virtual ground acceleration history and the convolution kernel, as shown in equations (29) and (30).

[0104] From step 4.1 to step 4.3, the time-step modal state recursion is transformed into an explicit causal discrete convolution between the virtual ground acceleration history and the modal convolution kernel, providing the basis for constructing the target response direct output kernel in step 5 and implementing FFT convolution in step 6.

[0105] Step 5: Based on the modal displacement convolution kernel and modal velocity convolution kernel obtained in Step 4, construct a direct output kernel for the target response, which is oriented towards the target node displacement and the basis response. Specifically: Step 5.1: Based on the modal displacement convolution kernels and modal velocity convolution kernels obtained in Step 4.2, combine them according to modal order to obtain the... The modal displacement convolution kernel vector and modal velocity convolution kernel vector at each time lag are shown in formulas (31) and (32).

[0106] In this embodiment, the modal displacement convolution kernel vector and modal velocity convolution kernel vector at each time lag each contain 65 modal components. Through this combination, the single-order modal convolution kernel can be extended into a vector form oriented towards multimodal superposition.

[0107] Step 5.2: Based on the modal displacement convolution kernel vector obtained in Step 5.1, the target node mode shape output relationship is pre-combined using the target node displacement selection matrix to obtain the first... The target node displacement at the time lag is directly output as the kernel, as shown in formula (33).

[0108] In this embodiment, the first node 1, the second node 2, the third node 3, and the fourth node 4 at different heights of the platform are selected as target nodes, and their X-direction displacement responses are output. Figures 4-7 The power spectral density of the X-direction displacement response evolution of the first node 1, the second node 2, the third node 3, and the fourth node 4, calculated based on the method of this invention, is presented. Figures 4-7It can be seen that the displacement responses of nodes 1, 2, 3, and 4 all exhibit significant time non-stationarity: the response spectral density gradually increases during the rising phase of the ground motion (0–10 s), remains at a high level during the stable phase of strong earthquakes (10–30 s), and gradually decreases after 30 s as the envelope function decays. Along the frequency direction, the displacement response energy is mainly concentrated in the low-frequency range of approximately 4–5 rad / s. Simultaneously, the amplitude of the displacement evolution power spectral density decreases sequentially from node 1 to node 4.

[0109] Step 5.3: Divide the complete mass matrix, damping matrix and stiffness matrix containing the constrained degrees of freedom into blocks according to the unconstrained degrees of freedom and constrained degrees of freedom, and obtain the block matrix as shown in formula (34) and formula (35).

[0110] In this embodiment, the six degrees of freedom of the bottom support node are used as constrained degrees of freedom, and the remaining dynamic degrees of freedom are used as free degrees of freedom. By dividing the complete mass matrix, damping matrix, and stiffness matrix into blocks, the mass coupling matrix, damping coupling matrix, and stiffness coupling matrix between the constrained degree-of-freedom rows and the free degree-of-freedom columns are obtained.

[0111] For the Rayleigh damped system obtained in step 1.4, the damping coupling matrix corresponding to the constrained degree-of-freedom row and the unconstrained degree-of-freedom column is shown in formula (36).

[0112] Step 5.4: Based on the block matrix obtained in step 5.3, the virtual support reaction vector is recovered from the dynamic equilibrium equation corresponding to the constrained degrees of freedom, as shown in formula (37).

[0113] In this embodiment, the relative displacement, relative velocity, and relative acceleration at the fixed support are all zero, but the constrained degree-of-freedom row still includes the contribution of the free degree-of-freedom motion to the support reaction force. Therefore, through the equilibrium equations of the constrained degree-of-freedom row, the virtual generalized displacement, virtual generalized velocity, virtual generalized acceleration, and the direct inertial term of the ground acceleration of the free degree-of-freedom are collectively converted into the support reaction force. This step avoids estimating the base response solely based on the equivalent restoring force of the free degree-of-freedom, improving the consistency between the base response calculation and the overall dynamic equilibrium relationship.

[0114] Step 5.5: Based on the virtual reaction force vectors at the constrained degrees of freedom obtained in Step 5.4, extract the virtual reaction forces and virtual reaction moments of each support node according to the order of the degrees of freedom of the support nodes, and synthesize them into the resultant force and resultant moment of the base with respect to the base reference point. The resultant force and resultant moment of the base are as shown in Equations (39) and (40).

[0115] In this embodiment, the base reference point is taken as the geometric center of the mud-surface pile head node coordinates. The base response includes the resultant base force in three directions and the resultant base moment in three directions, among which the base shear force in the X direction and the overturning moment about the Y-axis are the overall response quantities that are the focus of analysis in this embodiment.

[0116] Step 5.6: Based on the resultant force and resultant moment of the base obtained in step 5.5, construct the base response vector according to the fixed order of the resultant force of the base in the three directions first and the resultant moment of the base in the three directions last, as shown in formula (41).

[0117] In this embodiment, the base response vector consists of the resultant base forces in the X, Y, and Z directions and the resultant base moments about the X, Y, and Z directions. Figure 8 and Figure 9 The evolutionary power spectral densities of the base shear force in the X direction and the overturning moment about the Y axis are presented respectively. Figure 8 and Figure 9 It can be seen that the base response and nodal displacement response also exhibit obvious non-stationary evolution characteristics, and the variation of their spectral density over time is basically consistent with the envelope of the input ground motion intensity. From the frequency domain distribution, the main energy of the base shear force and overturning moment is concentrated in the low-frequency region, indicating that the overall base response is closely related to the low-order bending modes of the platform. The results show that the constraint reaction force recovery and base synthesis treatment in this invention can reflect the main frequency characteristics of the overall dynamic response of the jacket platform.

[0118] Step 5.7: Based on the virtual reaction force vector of constrained degrees of freedom and the base response vector obtained in steps 5.4 to 5.6, define the transformation matrices of modal acceleration, modal velocity, modal displacement and ground acceleration direct terms to the base response, as shown in formulas (42) to (45).

[0119] In this embodiment, the transformation matrix directly maps the modal space response to the basis response space, enabling the basis response to be pre-combined with the modal convolution kernel, without having to recover the complete physical coordinate response at each discrete frequency point and then calculate the support reaction force and the basis synthesis quantity item by item.

[0120] Step 5.8, based on the transformation matrix obtained in step 5.7 , , , The virtual generalized acceleration vector is eliminated using the modal dynamic equations obtained in step 2.5, so that the basis response is determined only by the virtual generalized displacement. Virtual generalized speed and virtual ground acceleration Define the modal damping diagonal matrix and the modal frequency squared diagonal matrix as shown in formulas (46) and (47).

[0121] In this embodiment, virtual generalized acceleration is represented as a combination of virtual generalized displacement, virtual generalized velocity and virtual ground acceleration, as shown in formula (48).

[0122] Substituting equation (48) into equation (41), we obtain the basis response that does not explicitly include virtual generalized acceleration. The basis response can be further written as three parts: modal displacement term, modal velocity term, and direct ground acceleration term, as shown in equation (49). This process avoids calculating and storing the virtual generalized acceleration time history separately within the frequency cycle, further reducing the storage and computational overhead of the basis response calculation.

[0123] Step 5.9, based on the modal displacement convolution kernel vector obtained in Step 5.1 and modal velocity convolution kernel vector and the basis response transformation matrix obtained in step 5.8 , , Construct the first The basis response at a time lag directly outputs the kernel, as shown in Equation (53).

[0124] In this embodiment, the direct output kernel of the basis response integrates the recovery of support reaction force, the synthesis of base resultant force and resultant moment, modal acceleration elimination, and modal convolution kernel into a single kernel sequence. For the first time lag term, the direct inertial term of ground acceleration is incorporated into the convolution kernel; for the remaining historical lag terms, only the historical contributions of the modal displacement and modal velocity terms are retained.

[0125] Step 5.10: Directly output the kernel based on the target node displacement obtained in Step 5.2. The basis response obtained in step 5.9 is directly output as the kernel. By vertically combining the two parallel output branches in a fixed order, we obtain the first... r The target response direct output kernels at each time lag are formed into a target response direct output kernel sequence according to the time lag order, as shown in formulas (54) and (55).

[0126] Therefore, both the target node displacement and the basis response can be obtained directly from the same target response output kernel.

[0127] This step is a crucial difference between this invention and conventional scale-overlay post-processing methods. Conventional methods typically require first recovering the complete physical displacement, velocity, and acceleration time histories at each discrete frequency point before further calculating the support reactions and base responses. In contrast, this invention pre-merges the target node displacement output relationship and the base response output relationship into the target response direct output kernel, enabling the target response to be directly output within the frequency cycle.

[0128] Step 6: Directly output the kernel based on the target response obtained in Step 5. The virtual ground acceleration at each discrete frequency point obtained in step 2 Under the condition of satisfying the linear convolution length requirement, fast Fourier transform, element-wise multiplication in the frequency domain, and inverse fast Fourier transform are performed to obtain the virtual response of the target node displacement and the virtual response of the basis within the effective time period; as follows: Step 6.1, based on the number of valid time points obtained in Step 4 The length of the Fast Fourier Transform is determined according to formula (56) and is not less than the length of the complete linear convolution.

[0129] In this embodiment, the effective time points are 8000, and the length of the complete linear convolution is 15999. To avoid aliasing caused by circular convolutions in the FFT calculation, the Fast Fourier Transform length is taken as an integer power of least two that is not less than the length of the complete linear convolution, i.e., 16384. This value ensures that the linear convolution result is not contaminated by circular convolutions within the effective time period.

[0130] Step 6.2, based on the virtual ground acceleration obtained in Step 2.2 Construct the first according to the effective time point sequence The effective time history vector of virtual ground acceleration corresponding to each discrete frequency point is shown in formula (57).

[0131] In this embodiment, for each discrete frequency point within the range of 0–15 rad / s, a virtual ground acceleration effective time history vector of length 8000 is constructed. This vector contains a three-segment intensity envelope function and a harmonic phase term corresponding to the discrete frequency point, thus reflecting the evolution characteristics of non-stationary seismic input in both time and frequency dimensions.

[0132] Step 6.3: Directly output the kernel sequence based on the target response obtained in step 5.10. And the virtual ground acceleration effective time history vector obtained in step 6.2 Zeros are padded at the ends of both time series to extend their time dimensions to the length of the Fast Fourier Transform determined in step 6.1. Then, the Fast Fourier Transform, element-wise multiplication in the frequency domain, and Inverse Fast Fourier Transform are performed sequentially along the time dimension. The required effective time segment is extracted from the Inverse Fast Fourier Transform result to obtain the first... The virtual time history matrix of the target response corresponding to each discrete circular frequency point is shown in formula (58).

[0133] In this embodiment, the FFT of the target response directly output kernel sequence only needs to be pre-calculated once. After entering the discrete frequency loop, the effective time history vector of the virtual ground acceleration at the current discrete frequency point is generated sequentially, its FFT is calculated, and element-wise multiplied with the target response directly output kernel spectrum. Then, the inverse FFT is performed, and the first 8000 effective time points are extracted. This converts the time-by-time historical accumulation into frequency domain multiplication. In this embodiment, the computing environment is a Windows 11 operating system, an Intel(R) i7-14700K 3.40GHz processor, 32GB of memory, and Python version 3.13.12, using single-process, single-threaded computation. The above hardware and software environment is only used to illustrate the computational effect of this embodiment and does not constitute a limitation on the implementation platform of this invention.

[0134] From steps 6.1 to 6.3, the target response is directly output as the kernel. With virtual ground acceleration effective time history vector The time-domain linear convolution between them is converted into frequency-domain multiplication, and the virtual time history of the target node displacement and the basis response is directly obtained for each discrete frequency point.

[0135] Step 7: Based on the target response virtual time history matrix obtained in Step 6, calculate the target response evolution power spectral density matrix and time-varying variance matrix sequentially, and complete the statistical output. Details are as follows: Step 7.1, based on the target response virtual time history matrix obtained in step 6.3 Calculate the first using formula (59) h The power spectral density matrix of the target response evolution corresponding to each discrete circular frequency point.

[0136] Equation (59) gives the time history matrix of the self-evolving power spectral density of the target response in the th... h The values ​​at discrete circular frequency points. Let Represents the change in frequency with continuous circular frequency The changing target response self-evolves the power spectral density time history matrix, then For the matrix function in the th h Discrete circular frequency points Discrete values ​​at a given point.

[0137] In this embodiment, the target response evolution power spectral density matrix includes the auto-power spectral density of the target node's X-direction displacement and the evolution power spectral density matrix of the basis response. The matrix simultaneously contains auto-spectral terms of each target response component and cross-spectral terms between different target response components; the correlation terms generated by modal superposition can be preserved through complex conjugate outer product.

[0138] Step 7.2: Based on the target response self-evolution power spectral density time history matrix obtained in step 7.1, integrate along discrete frequency points to obtain the target response time-varying variance matrix, as shown in formula (60).

[0139] In this embodiment, trapezoidal integral weights are used for frequency integration. By weighted summing of the evolved power spectral density at each discrete frequency point in the range of 0–15 rad / s, the time-varying variance of the target response at each effective time point is obtained.

[0140] Step 7.3: Based on the target response time-varying variance vector obtained in step 7.2, take the square root of each element to obtain the time-varying standard deviation of the target nodal displacement and the basis response.

[0141] In this embodiment, to verify the accuracy and computational efficiency of the method of the present invention, the PEM / EPIM-FFT calculation method of the present invention is compared with the Newmark average acceleration method and the HPD-S method using the same jacket model, the same modal truncation order, the same non-stationary seismic input, the same time step, and the same frequency discretization parameters. Figure 10 The time-varying standard deviations of the X-direction displacements for the first node 1, the second node 2, the third node 3, and the fourth node 4 are given. Figure 10 It can be seen that the curves obtained by the PEM / EPIM-FFT calculation method, the Newmark average acceleration method, and the HPD-S method of this invention basically overlap throughout the entire analysis period. During the stable phase of the strong earthquake, the standard deviations of the X-direction displacements of the first node 1, the second node 2, the third node 3, and the fourth node 4 are 4.9 cm, 4.3 cm, 3.5 cm, and 2.7 cm, respectively, showing a decreasing trend with decreasing node elevation. This result is consistent with... Figures 4-8 The consistent amplitude distribution of the power spectral density of the displacement response along the platform height indicates that the method of the present invention can accurately reflect the statistical response of non-stationary displacement at different height positions of the jacket platform.

[0142] Figure 11 and Figure 12 The time-varying standard deviations of the base shear force in the X direction and the overturning moment about the Y axis are given respectively. Figure 11 and Figure 12 It can be seen that the standard deviation curves of the base response obtained by the PEM / EPIM-FFT calculation method, the Newmark average acceleration method, and the HPD-S method of this invention are basically consistent. The standard deviation of the base shear force in the X direction is approximately 769 kN in the stable phase of a strong earthquake, and the standard deviation of the overturning moment around the Y axis is approximately 45800 kNm. The consistency of the results of the three methods indicates that the target response constructed by this invention directly outputs the recovery relationship of the undamaged constraint reaction force of the core, and can accurately output the non-stationary statistical response of the overall base reaction force and the resultant moment of the base.

[0143] In terms of computational efficiency, all three methods employ single-process, single-threaded computation. The timing scope includes pre-calculation processes such as the state transition matrix, load integral vector, and target response direct output kernel, as well as the core response calculation for all virtual excitation frequency points. The computation time is as follows: Figure 13 As shown, the total computation time of the Newmark average acceleration method is 14.86 s, the total computation time of the HPD-S method is 23.17 s, and the total computation time of the PEM / EPIM-FFT calculation method of this invention is 4.95 s. Therefore, under the same computational conditions in this embodiment, the total computation time of the PEM / EPIM-FFT calculation method of this invention is reduced by 66.7% compared to the Newmark average acceleration method and by 78.6% compared to the HPD-S method.

[0144] Furthermore, using the PEM / EPIM-FFT calculation results of the present invention as a comparison benchmark, the maximum relative error of the time-varying standard deviation of nodal displacement obtained by the Newmark average acceleration method is 0.103%, and the maximum relative error of the time-varying standard deviation of nodal displacement obtained by the HPD-S method is 0.153%. This indicates that the PEM / EPIM-FFT calculation method of the present invention can significantly reduce the calculation cost of multi-frequency, long-time-history non-stationary random seismic response analysis while maintaining the accuracy of nodal displacement and base response calculations.

[0145] This embodiment also provides a PEM / EPIM-FFT calculation system for the non-stationary random seismic response of a jacket platform, the system structure of which is as follows: Figure 14 As shown, it includes a model input and dynamic modeling module, a non-stationary virtual excitation construction module, a mode decomposition and decoupling module, an EPIM pre-computation module, a mode convolution kernel generation module, a target response direct output kernel construction module, an FFT response calculation module, and a statistical output module.

[0146] The model input and dynamic modeling module is used to read the structural mass matrix, stiffness matrix, nodal coordinates, constraint degrees of freedom, seismic input direction, time discretization parameters, frequency discretization parameters, and non-stationary ground motion parameters of the finite element model of the jacket platform, and to transmit these parameters to the non-stationary virtual excitation construction module and the modal decomposition module. In this embodiment, the model input module reads the finite element model data of the four-legged jacket platform, the X-direction seismic input direction, the 80s seismic duration, the 0.01s time step, the 0-15rad / s frequency integration range, and the 0.01rad / s frequency step.

[0147] A non-stationary virtual excitation construction module is used to construct a deterministic intensity envelope function, a stationary base process power spectral density, and a modulated harmonic virtual ground acceleration based on the non-stationary ground motion parameters. In this embodiment, the module constructs a three-segment intensity envelope function and a Kanai-Tajimi spectrum, and generates a corresponding modulated harmonic virtual ground acceleration for each discrete frequency point.

[0148] The modal decomposition and decoupling module is used to solve for the mode shape matrix and natural circular frequency based on the total mass matrix and stiffness matrix of the structure with hydrodynamic added mass, and to generate virtual generalized coordinate equations that are decoupled step by step. In this embodiment, the module retains 65 modes.

[0149] The EPIM pre-calculation module is used to calculate and store the modal state transition matrix and load integral vector based on the virtual generalized coordinate equations of successive decoupling. In this embodiment, the module calculates the modal state transition matrix and load integral vector for each of the 65 modes and moves them to the frequency out-of-cycle pre-calculation.

[0150] The modal convolution kernel generation module is used to generate modal displacement convolution kernels and modal velocity convolution kernels based on the modal state transition matrix, load integral vector, and mode shape participation coefficients. In this embodiment, the module generates modal displacement convolution kernels and modal velocity convolution kernels corresponding to 8000 valid time points.

[0151] The target response direct output kernel construction module is used to construct the target response direct output kernel based on the modal displacement convolution kernel, modal velocity convolution kernel, target nodal mode shape output relationship, constraint reaction force recovery relationship, and basis response synthesis relationship. In this embodiment, the module pre-combines the nodal X-direction displacement output relationship, support reaction force recovery relationship, and basis response synthesis relationship into the target response direct output kernel.

[0152] The FFT response calculation module performs Fast Fourier Transform, frequency domain element-wise multiplication, Inverse Fast Fourier Transform, and effective time-time truncation on the target response direct output kernel and the effective time-history vector of the virtual ground acceleration, to obtain the target node displacement virtual response and the basis virtual response. In this embodiment, the FFT length is 16384.

[0153] The statistical output module is used to calculate the evolving power spectral density of the target response based on the conjugate outer product of the virtual response of the target node displacement and the virtual response of the base, and to output the time-varying standard deviation of the target node displacement and the base response by integrating along the frequency. In this embodiment, the module outputs the evolving power spectral density and time-varying standard deviation of the node X-direction displacement, the X-direction base shear force, and the overturning moment about the Y-axis.

[0154] This embodiment also provides an electronic device, which includes a processor and a memory. The memory stores a computer program, which, when executed by the processor, implements the above-described PEM / EPIM-FFT calculation method.

[0155] This embodiment also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the above-described PEM / EPIM-FFT calculation method.

[0156] The embodiments described above are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A PEM / EPIM-FFT calculation method for the non-stationary random seismic response of a jacket platform, characterized in that, The calculation method includes the following steps: Step 1: Obtain the structure, environment, seismic input, discrete parameters, and target response parameters of the jacket platform; calculate the hydrodynamic added mass for the members below the water surface, and sequentially form the total structural mass matrix, Rayleigh damping matrix, and structural relative motion equations to obtain the basic dynamic model; Step 2: Based on the fundamental dynamic model, construct the non-stationary virtual ground acceleration at each discrete frequency point, solve the mass-normalized mode shape and establish the modal coordinate transformation to obtain the virtual generalized coordinate equations that are decoupled step by step. Step 3: Based on the virtual generalized coordinate equation obtained in Step 2, establish the first-order modal state space equation, and use the explicit fine integration method to pre-calculate the modal state transition matrix and load integral vector to obtain the reusable state transition quantities outside the frequency cycle. Step 4: Based on the modal state transition matrix and load integral vector obtained in Step 3, the modal state response is recursively expanded under zero initial conditions, and the modal displacement convolution kernel, modal velocity convolution kernel and causal discrete convolution expression are obtained in sequence. Step 5: Based on the modal displacement convolution kernel and modal velocity convolution kernel obtained in Step 4, construct a target response direct output kernel oriented towards the target node displacement and basis response; Step 6: Based on the target response obtained in Step 5, directly output the kernel and the non-stationary virtual ground acceleration obtained in Step 2. Under the condition of satisfying the linear convolution length requirement, perform fast Fourier transform, frequency domain element-wise multiplication and inverse fast Fourier transform to obtain the target node displacement virtual response and the basis virtual response within the effective time period. Step 7: Based on the target response virtual time history matrix obtained in Step 6, calculate the target response evolution power spectral density matrix and time-varying variance matrix in sequence, and complete the statistical output.

2. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 1, characterized in that, Step 1 specifically involves: Step 1.1: Obtain the complete structural mass matrix, complete stiffness matrix, node coordinates, member connection relationships, member outer diameter, member length, immersion range, constrained degrees of freedom, seismic input direction, structural damping ratio, time discretization parameters, frequency discretization parameters, non-stationary ground motion parameters, target nodes and their displacement directions, support nodes and base reference points of the finite element model of the jacket platform. The finite element model of the jacket platform is divided into unconstrained degrees of freedom and constrained degrees of freedom. The structural dynamic response is solved on the unconstrained degrees of freedom, and the support reaction force at the constrained position is recovered according to the dynamic equilibrium equations corresponding to the constrained degrees of freedom. Step 1.2: Based on the node coordinates, member connection relationships, member outer diameters, member lengths, and immersion intervals obtained in Step 1.1, the inertial force per unit length of the submerged member is calculated using the Morison equation's inertial force term, yielding the equivalent nodal hydrodynamic mass of each member element; e The hydrodynamic inertial force per unit length of a circular tube element is expressed as: （1） in, For the first e Each rod element in depth coordinates z and time t Dynamic inertial force per unit length at a given point; subscript It represents water; e Number the member units; The hydrodynamic inertia coefficient; The density of seawater; Pi; No. e The outer diameter of each bar element; For the first e Each rod element in depth coordinates z Relative acceleration at the point; The first e When the hydrodynamic added mass of a single member element is evenly distributed to the two end nodes, the equivalent hydrodynamic added mass of a single end node is expressed as: （2） in, For the first e The hydrodynamic additional mass of each member element is allocated to a single-end node; For the first e Length of each bar element; Step 1.3: Based on the equivalent nodal hydrodynamic mass of each member element, assemble them into a hydrodynamic mass matrix according to the nodal degrees of freedom. Superimpose this matrix with the structural mass matrix to obtain the total structural mass matrix used for unconstrained degree-of-freedom dynamic solutions. （3） in, The total mass matrix of the structure; This is the structural mass matrix corresponding to the unconstrained degrees of freedom; This represents the hydrodynamic added mass matrix corresponding to the unconstrained degrees of freedom; subscript Represents the structural entity; Step 1.4: Based on the complete stiffness matrix and structural damping ratio obtained in Step 1.1, and the total structural mass matrix obtained in Step 1.3, first solve the undamped characteristic equation to obtain the first and second natural circular frequencies used for damping calibration, and then construct the damping matrix using Rayleigh damping: （4） （5） （6） in, This is the damping matrix corresponding to the unconstrained degrees of freedom; This is the stiffness matrix corresponding to the unconstrained degrees of freedom; Rayleigh's mass proportional damping coefficient; This refers to the Rayleigh stiffness proportional damping coefficient; The structural damping ratio; The first natural circular frequency of the structure; This is the second-order natural circular frequency; When calculating the damping matrix, simultaneously follow the same and Construct a complete damping matrix containing the constrained degrees of freedom; Step 1.5: Based on the total structural mass matrix obtained in Step 1.3, the damping matrix and stiffness matrix obtained in Step 1.4, and the seismic input direction obtained in Step 1.1, establish the structural relative motion equations for the hydrodynamically added mass: （7） in, The relative displacement vector of the structural nodes; It is a relative velocity vector; It is a relative acceleration vector; Input the direction vector for the earthquake; Ground acceleration time history; subscript Indicates ground sports markings; Steps 1.1 to 1.5 ultimately form the basic dynamic model.

3. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 2, characterized in that, Step 2 specifically involves: Step 2.1: Based on the non-stationary ground motion parameters obtained in Step 1.1, the ground acceleration is expressed as the product of a deterministic intensity envelope function and a zero-mean stationary Gaussian process, yielding the evolutionary power spectral density of the ground acceleration: （8） （9） in, The deterministic intensity envelope function; It is a zero-mean stationary Gaussian process; for The evolution of power spectral density; For the circular frequency variable; Step 2.2: Based on the evolved power spectral density and deterministic intensity envelope function, and based on the frequency discrete parameters obtained in Step 1.1, the virtual excitation method is used to construct the first... h Modulated harmonic virtual ground acceleration at discrete frequency points: （10） in, For the first h Virtual ground acceleration corresponding to each discrete frequency point; h Number the discrete frequency points. , This represents the total number of discrete frequency points. For the first h A discrete circular frequency; i is the imaginary unit; Step 2.3: Based on the total structural mass matrix and stiffness matrix obtained in Step 1, solve the characteristic equation of undamped free vibration considering the hydrodynamic added mass, and obtain the mode shape matrix and the natural circular frequencies of each order: （11） （12） （13） in, For the first Mode shape vector; Modal order numbering, ; To preserve the modal order; For the first The natural circular frequency of the first order; To preserve the mode shape matrix of the modal composition; T Indicates matrix transpose; for An identity matrix of order 1; Step 2.4, based on the mode shape matrix obtained in Step 2.3, the first... By projecting the virtual displacement response corresponding to each discrete frequency point onto modal coordinates, the relationship between the virtual physical displacement and the virtual generalized coordinates can be obtained: （14） in, For the first The virtual physical displacement response vector corresponding to each discrete frequency point; It is a virtual generalized coordinate vector composed of all preserved modes; For the first Virtual generalized coordinates; Step 2.5: Based on the virtual ground acceleration obtained in Step 2.2 and the modal coordinate transformation obtained in Step 2.4, decouple the structural relative motion equation (7) in Step 1.5 to obtain the first... Virtual generalized coordinate equations for step-by-step decoupling: （15） in, For the first Virtual generalized velocity; For the first Order virtual generalized acceleration; For the first First-order modal damping ratio; For the first The mode participation factor is expressed as: （16） Steps 2.1 to 2.5 transform the multi-degree-of-freedom nonstationary random seismic response problem into a deterministic complex-valued single-degree-of-freedom response problem at different discrete frequency points and different modal orders.

4. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 3, characterized in that, Step 3 specifically involves: Step 3.1, based on the virtual generalized coordinate equations obtained in Step 2.5, define the modal state vector consisting of virtual generalized displacement and virtual generalized velocity: （17） The virtual generalized coordinate equations are then rewritten as first-order modal state-space equations: （18） in, For the first First-order modal state vector; The first-order time derivative of the modal state vector; For the first The first-order modal state matrix; This is the load input vector; Step 3.2: Based on the time discretization parameters obtained in Step 1.1, discretize the time interval into... The time step is Explicit fine integration is performed on the first-order modal state-space equations obtained in step 3.1 to obtain the single-step state transition relations: （21） in, For the first A discrete time point; Number the time steps. ; To count the number of valid time points after removing the initial time, the index is... Indicates the effective length; For time step; For the first The modal state transition matrix of order ; For the first The load integral vectors of order are respectively expressed as: （22） （23） in, For the time variable in the load integration process.

5. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 4, characterized in that, Step 4 specifically involves: Step 4.1, under the conditions of zero initial displacement and zero initial velocity, recursively expand the single-step state transition relationship in Step 3.2 to obtain the... The causal discrete convolution expression for the first-order modal state response: （24） in, For the first h A discrete circular frequency point; In the first h Discrete circular frequency points Under the corresponding virtual ground acceleration, the first j The first mode in the first discrete time points The modal state vector; Number historical time points; For the first A historical point in time; The modal state transition matrix Power of; For the first h Discrete circular frequency points The corresponding virtual ground acceleration is at the 1st Historical moments The complex value amplitude; Step 4.2, based on the modal state response obtained in Step 4.1, define... The current time point and historical time points The discrete time lag positions between are determined, and the first and second components of the state vector are extracted using the displacement selection vector and velocity selection vector, respectively, to obtain the first... The first mode in the first Modal displacement convolution kernel and modal velocity convolution kernel at each time lag: （25） （26） in, Number the time lag; For the first The first mode in the first Modal displacement convolution kernel at a time lag; For the first The first mode in the first Modal velocity convolution kernels at time lags; The modal state transition matrix Power of; Choose a vector for the displacement; Choose a vector for velocity; Step 4.3: Based on the modal displacement convolution kernel and modal velocity convolution kernel obtained in Step 4.2, the first... The virtual generalized displacement and virtual generalized velocity are represented as a causal discrete convolution between the virtual ground acceleration history and the convolution kernel: （29） （30） in, In the first h Under the action of virtual ground acceleration corresponding to the discrete circular frequency, the first j The first mode at the current time point Virtual generalized displacement; For the first j The first mode in the first r Modal displacement convolution kernel at each time lag position; In the first h Under the action of virtual ground acceleration corresponding to the discrete circular frequency, the first j The first mode at the current time point Virtual generalized velocity; For the first j The first mode in the first r Modal velocity convolution kernels at time lag positions.

6. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 5, characterized in that, Step 5 specifically involves: Step 5.1: Based on the modal displacement convolution kernels and modal velocity convolution kernels obtained in Step 4.2, combine them according to modal order to obtain the... Modal displacement convolution kernel vector and modal velocity convolution kernel vector at each time lag: （31） （32） in, For the first A column vector consisting of all preserved modal shift convolution kernels at a time lag; For the first A column vector consisting of all preserved modality velocity convolution kernels at each time lag; For the first-order mode in the... Modal displacement convolution kernel at a time lag; For the second-order mode in the... Modal displacement convolution kernel at a time lag; For the first The first mode in the first Modal displacement convolution kernel at a time lag; For the first-order mode in the... Modal velocity convolution kernels at time lags; For the second-order mode in the... Modal displacement convolution kernel at a time lag; For the first The first mode in the first Modal velocity convolution kernels at time lags; Step 5.2: Based on the modal displacement convolution kernel vector obtained in Step 5.1, the target node mode shape output relationship is pre-combined using the target node displacement selection matrix to obtain the first... The target node displacement at the time lag is directly output as the kernel: （33） in, For the first The target node displacement at the time lag point is directly output as the kernel; A target node displacement selection matrix is ​​used to extract pre-specified target nodes and displacement directions from unconstrained physical displacements; subscripts Indicates displacement output identifier; Step 5.3: Divide the complete mass matrix, damping matrix, and stiffness matrix containing the constrained degrees of freedom into blocks according to the unconstrained and constrained degrees of freedom, to obtain the block matrix: （34） （35） in, The complete mass matrix containing the constrained degrees of freedom; This is the complete stiffness matrix including constrained degrees of freedom; the subscript f indicates the unconstrained degree of freedom number; the subscript c indicates the constrained degree of freedom number. This is the mass coupling matrix corresponding to the constrained degree-of-freedom rows and the unconstrained degree-of-freedom columns; This is the mass coupling matrix corresponding to the unconstrained degree-of-freedom rows and constrained degree-of-freedom columns; This is the mass coupling matrix corresponding to the unconstrained degree-of-freedom rows and columns; This is the mass coupling matrix corresponding to the constrained degree-of-freedom rows and columns; This is the stiffness coupling matrix corresponding to the constrained degree-of-freedom rows and the unconstrained degree-of-freedom columns; This is the stiffness coupling matrix corresponding to the unconstrained degree-of-freedom rows and constrained degree-of-freedom columns; This is the stiffness coupling matrix corresponding to the unconstrained degree-of-freedom rows and columns; This is the stiffness coupling matrix corresponding to the constrained degree-of-freedom rows and columns; For the Rayleigh damped system obtained in step 1.4, the damping coupling matrix corresponding to the constrained degree-of-freedom row and the unconstrained degree-of-freedom column is: （36） in, This is the damping coupling matrix corresponding to the constrained degree-of-freedom rows and unconstrained degree-of-freedom columns in the complete damping matrix; Step 5.4: Based on the block matrix obtained in Step 5.3, recover the virtual reaction force vector of the constrained degrees of freedom from the dynamic equilibrium equations corresponding to the constrained degrees of freedom: （37） in, This refers to the virtual reaction force vector at the constrained degrees of freedom; A virtual generalized acceleration vector that preserves all modes; The virtual generalized velocity vector that preserves all modes; For all modes preserved, the virtual generalized displacement vector; Let be the direct inertial influence vector of ground acceleration on constraint reaction force, expressed as: （38） in, This represents the seismic input direction vector corresponding to the unconstrained degrees of freedom. The seismic input direction vector corresponding to the constrained degrees of freedom; Step 5.5: Based on the virtual reaction force vectors at the constrained degrees of freedom obtained in Step 5.4, extract the virtual reaction forces and virtual reaction moments of each support node according to the order of the support node degrees of freedom, and synthesize them into the resultant force and resultant moment of the base with respect to the base reference point; then the resultant force and resultant moment of the base are expressed as: （39） （40） in, The resultant force vector of the basis; For the first Virtual reaction force vectors of each support node; Number the support nodes; This represents the total number of support nodes; The resultant moment vector of the basis forces; For the first The virtual reaction moment vector of each support node; Represents the cross product of vectors; For the first Position vectors of each support node relative to the base reference point; subscript Indicates the basis response identifier; Step 5.6: Based on the resultant force and moment of the base obtained in Step 5.5, construct the base response vector according to a fixed order of resultant force in the three directions first, followed by resultant moment in the three directions: （41） in, The basis virtual response vector; , and These are the resultant base forces in the X, Y, and Z directions, respectively; , and These are the resultant moments of the base forces about the X, Y, and Z directions, respectively. The basis response synthesis matrix is ​​used to map the virtual reaction force vectors of constrained degrees of freedom to the basis response vectors. Step 5.7: Based on the constrained degrees of freedom virtual reaction force vector and basis response vector obtained in steps 5.4 to 5.6, define the transformation matrices of modal acceleration, modal velocity, modal displacement, and ground acceleration direct terms to the basis response, respectively: （42） （43） （44） （45） in, This is the transformation matrix from modal acceleration to basis response; This is the transformation matrix from modal velocity to basis response; This is the transformation matrix from modal displacement to basis response; This is the transformation vector from the direct term of ground acceleration to the basis response; Step 5.8, based on the transformation matrix obtained in step 5.7 , , , The virtual generalized acceleration vector is eliminated using the modal dynamic equations obtained in step 2.5, so that the basis response is determined only by the virtual generalized displacement. Virtual generalized speed and virtual ground acceleration Define the modal damping diagonal matrix and the modal frequency squared diagonal matrix: （46） （47） in, This is the diagonal matrix of modal damping; This indicates that a diagonal matrix is ​​constructed from the elements within the parentheses; It is a diagonal matrix of modal frequencies squared; The damping ratio for the first-order mode; For the first First-order modal damping ratio; This is the first-order natural circular frequency; For the first The natural circular frequency of the first order; The virtual generalized acceleration vector, composed of the retained modal dynamic equations, is represented as follows: （48） in, This is the vector of modal participation coefficients. This is the participation factor for the first-order mode shape. This is the participation factor for the second-order mode shape. For the first Mode participation factor; Substituting equation (48) into equation (41), we obtain the basis response that does not explicitly include virtual generalized acceleration: （49） in, This is the basis response transformation matrix corresponding to the modal displacement term; This is the basis response transformation matrix corresponding to the modal velocity term; This is the basis response transformation vector corresponding to the direct term of the virtual ground acceleration; Step 5.9, based on the modal displacement convolution kernel vector obtained in Step 5.1 and modal velocity convolution kernel vector and the basis response transformation matrix obtained in step 5.8 , , Construct the first The basis response at each time lag directly outputs the kernel: （53） in, For the first The basis response at a time lag directly outputs the kernel; For the Kronecker function, when hour ,when hour ; Step 5.10: Directly output the kernel based on the target node displacement obtained in Step 5.

2. The basis response obtained in step 5.9 is directly output as the kernel. By vertically combining the two parallel output branches in a fixed order, we obtain the first... r The target response direct output kernels at each time lag are formed into a sequence of target response direct output kernels in order of time lag: （54） （55） in, For the first The target response at the time lag point is directly output as the kernel; The kernel sequence is directly output to arrange the target responses in order of time lag. The kernel is directly output for the target response at the first time lag. The kernel is directly output as the target response at the second time lag. For the first The target response at a time lag is directly output as the kernel; subscript Indicates the target response identifier; The target node displacement and basis response are obtained by directly outputting the kernel from the same target response.

7. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 6, characterized in that, Step 6 specifically involves: Step 6.1, based on the number of valid time points obtained in Step 4 Determine the Fast Fourier Transform length that is not less than the length of the complete linear convolution: （56） in, The length of the Fast Fourier Transform; For two lengths The length of the sequence after performing a full linear convolution; Indicates rounding up to the nearest integer; Indicates the Fast Fourier Transform identifier; Step 6.2, based on the virtual ground acceleration obtained in Step 2.2 Construct the first according to the effective time point sequence The effective time history vector of virtual ground acceleration corresponding to each discrete frequency point: （57） in, For the first The effective time history vector of virtual ground acceleration corresponding to each discrete frequency point; For the first The complex amplitude of the virtual ground acceleration corresponding to each discrete frequency point at the first effective discrete time point; For the first The complex amplitude of the virtual ground acceleration corresponding to each discrete frequency point at the second effective discrete time point; For the first The virtual ground acceleration corresponding to the discrete frequency point, at the th discrete frequency point, is... The complex-valued amplitude at each effective discrete time point; Step 6.3: Directly output the kernel sequence based on the target response obtained in step 5.

10. And the virtual ground acceleration effective time history vector obtained in step 6.2 Zeros are padded at the ends of both time series to extend their time dimensions to the length of the Fast Fourier Transform determined in step 6.1; Fast Fourier Transform, element-wise multiplication in the frequency domain, and Inverse Fast Fourier Transform are then performed sequentially along the time dimension, and the required effective time segment is extracted from the Inverse Fast Fourier Transform result to obtain the first... The virtual time history matrix of the target response corresponding to each discrete circular frequency point: （58） in, For the first The target response virtual time history matrix corresponds to each discrete circular frequency point, with each row corresponding to a target response component and each column corresponding to an effective time point. Indicates length is Fast Fourier Transform; Indicates length is The inverse fast Fourier transform; This indicates element-wise multiplication in the frequency domain; Indicates the truncation before One effective time point; From steps 6.1 to 6.3, the target response is directly output as the kernel. With virtual ground acceleration effective time history vector The time-domain linear convolution between them is converted into frequency-domain multiplication, and the virtual time history of the target node displacement and the basis response is directly obtained for each discrete frequency point.

8. The PEM / EPIM-FFT calculation method for the non-stationary random seismic response of the jacket platform according to claim 7, characterized in that, Step 7 specifically involves: Step 7.1, based on the target response virtual time history matrix obtained in step 6.3 Calculate the first h The power spectral density matrix of the target response evolution corresponding to each discrete circular frequency point: （59） in, For the first h The time history matrix of the target response self-evolutionary power spectral density corresponding to each discrete circular frequency point; superscript Indicates complex conjugation; Step 7.2: Based on the time history matrix of the self-evolving power spectral density of the target response, integrate along discrete frequency points to obtain the time-varying variance matrix of the target response: （60） in, The time-varying variance matrix of the target response; For continuous circular frequency variable The changing target response self-evolves power spectral density matrix; The starting angular frequency of the frequency discrete interval; The terminating angular frequency of the frequency discrete interval; Step 7.3: Based on the target response time-varying variance vector obtained in step 7.2, take the square root of each element to obtain the time-varying standard deviation of the target nodal displacement and the basis response.

9. A PEM / EPIM-FFT calculation system for the non-stationary random seismic response of a jacket platform, characterized in that, The PEM / EPIM-FFT calculation method described in any one of claims 1-8 is implemented through a PEM / EPIM-FFT calculation system, including a model input and dynamic modeling module, a non-stationary virtual excitation construction module, a mode decomposition and decoupling module, an EPIM pre-computation module, a mode convolution kernel generation module, a target response direct output kernel construction module, an FFT response calculation module, and a statistical output module.

10. The PEM / EPIM-FFT calculation system for the non-stationary random seismic response of the jacket platform according to claim 9, characterized in that, Specifically: The model input and dynamic modeling module is used to acquire the structural mass matrix, stiffness matrix, nodal coordinates, constraint degrees of freedom, structural damping ratio, seismic input direction, time discretization parameters, frequency discretization parameters, target response parameters, and non-stationary ground motion parameters of the finite element model of the jacket platform. It sequentially calculates the hydrodynamic added mass, the total mass matrix of the assembled structure, constructs the Rayleigh damping matrix, and establishes the structural relative motion equations. The basic dynamic model is then transferred to the modal decomposition and decoupling module and the target response direct output kernel construction module, and the non-stationary ground motion parameters are transferred to the non-stationary virtual excitation construction module. The non-stationary virtual excitation construction module is used to construct a deterministic intensity envelope function and a stationary base process power spectral density based on non-stationary ground motion parameters, and to construct a modulated harmonic virtual ground acceleration for each discrete frequency point, and to transfer the virtual ground acceleration to the mode decomposition and decoupling module and the FFT response calculation module. The modal decomposition and decoupling module is used to solve the mass-normalized mode shape matrix and natural circular frequencies of each order based on the total mass matrix and stiffness matrix of the structure, and to combine the virtual ground acceleration with the modal coordinate transformation to generate virtual generalized coordinate equations that are decoupled step by step; the modal parameters and mode participation coefficients are passed to the EPIM pre-calculation module, the modal convolution kernel generation module and the target response direct output kernel construction module; The EPIM pre-computation module is used to calculate and store the modal state transition matrix and load integral vector of each order based on the virtual generalized coordinate equation and time step of successive decoupling; and to pass the modal state transition matrix and load integral vector to the modal convolution kernel generation module. The modal convolution kernel generation module is used to generate modal displacement convolution kernels and modal velocity convolution kernels in time lag order based on the modal state transition matrix, load integral vector, and mode participation coefficient; and to pass the modal convolution kernels to the target response direct output kernel construction module. The target response direct output kernel construction module is used to construct the target node displacement direct output kernel and the basis response direct output kernel, and merge the two into a target response direct output kernel sequence according to a fixed response order; The target response is directly output as a kernel sequence and passed to the FFT response calculation module. The FFT response calculation module is used to obtain the target response virtual time history matrix and transmit the target response virtual time history matrix to the statistical output module; The statistical output module is used to calculate the power spectral density matrix of the target response evolution based on the complex conjugate outer product of the virtual vector of the target response at each discrete frequency point, perform numerical integration along the circular frequency direction to obtain the time-varying variance vector of the target response, and output the time-varying standard deviation vector of the target node displacement and the basis response by taking the square root of each element.

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