Topology optimization method for marine engineering structures under continuous iterative random excitation based on analysis and design

Abstract

This application discloses a method for topological optimization of marine engineering structures under continuous iterative random excitation based on analysis and design. The root mean square of the dynamic response displacement at a specified key position under complex random excitation is taken as the objective function. In order to efficiently solve the random response of structural dynamics, a virtual excitation method is adopted. In order to overcome the huge computational cost of repeated modal solution in the dynamic optimization of structural random response, a continuous iterative random dynamic response optimization strategy based on analysis and design is proposed. Based on the adjoint method, the efficient optimization sensitivity of the random dynamic response of the structure is derived, and an optimization algorithm based on the moving asymptote method is adopted to solve the optimization problem. Optimization examples show that the proposed virtual excitation method based on analysis and design and the traditional double-loop virtual excitation method can obtain almost the same optimized design on the basis of significant computational efficiency advantages.

Description

Technical Field

[0001] The present invention belongs to the field of structure and multidisciplinary optimization design, and relates to a topology optimization and design method for an offshore wind power foundation structure. Background Art

[0002] The offshore wind power foundation structure is a fixed foundation structure widely used in marine engineering, mainly used to support facilities such as offshore wind farms. Its main body is composed of steel hollow columns and vertical and horizontal bars connecting the columns, forming a space frame structure. It is fixed to the seabed by steel pipe piles, has high rigidity, and can effectively resist natural loads such as waves, currents, and wind. The jacket foundation structure in the wind power foundation is usually used in sea areas with a water depth of 20 to 50 meters, especially in areas with a water depth of more than 30 meters. As the development of the ocean advances into the deep sea, the scope of use and structural form of the jacket foundation are also constantly improving. As a powerful design tool, topology optimization has been widely used in various fields such as mechanical structure and aerospace. In terms of offshore platform foundations, topology optimization has also carried out many optimization designs to form some advantageous structures.

[0003] However, previous topology optimization of offshore wind turbine foundations has mostly been conducted using static loads. However, offshore wind turbine foundations are subject to natural random loads such as waves and wind during actual operation. In the topology optimization of large-scale structures, dynamic optimization often results in excessively long optimization times and enormous computational complexity due to various factors, such as the solution method. This results in a very limited design space and unsatisfactory optimization results. Summary of the Invention

[0004] The purpose of the present invention is to provide a topology optimization method that effectively reduces the calculation time to address at least one of the shortcomings of large-scale structural topology optimization under random loads, such as long calculation time and low calculation efficiency.

[0005] To achieve the above objectives, some embodiments of the present invention propose a method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design, which includes the steps of: establishing a dynamic model of the marine engineering structure with N degrees of freedom, wherein the dynamic model includes a plurality of units; based on the dynamic model under random excitation obtained by using a virtual excitation method, the root mean square of the random displacement response of the rth degree of freedom in the N degrees of freedom is regarded as the objective function, and obtaining a topology optimization formula of the dynamic model:

[0006] ;

[0007] in, is the frequency of external excitation scope; is the design variable of the e-th unit; is the volume of the e-th unit; the constraints set by the optimization formula are: the total volume must be less than the set volume requirement; the root mean square of the objective function Expressed as: and Under the external excitation within the range, the power spectrum density The integral is obtained to measure the vibration amplitude of the marine engineering structure under the random excitation; V is the established volume requirement; f( t ) is non-zero D dimensional fixed random vector, whose autopower spectrum density is dimensional matrix; R is the unit transformation matrix, size is ;

[0008] Given the initial design variable values ​​for each of the units , solving the unit stiffness matrix and the unit mass matrix according to the initial design variable values, and generating the overall stiffness matrix K, the overall mass matrix M and the overall damping matrix C of the dynamic model according to the degree of freedom number and coordinate transformation;

[0009] solving the modal characteristics of the dynamic model using a continuous iterative method based on analysis and design, wherein the modal characteristics include natural modes and natural frequencies;

[0010] Substituting the obtained natural mode and natural frequency into the formula Among them, represents the modal vector under the nth mode, for The transposed matrix of represents the unit transformation vector, represents the kth vector obtained by Cholesky decomposition, represents the frequency domain transfer function under the nth mode, The complex exponential form of pseudo-harmonic excitation is used to solve the overall displacement response of the marine engineering structure under the kth pseudo-harmonic The displacement response of the rth degree of freedom of the marine engineering structure under the kth pseudo-harmonic , using the formula Get the displacement power spectrum density value of the rth degree of freedom , and then calculate the root mean square of the objective function to achieve the response solution to the random excitation;

[0011] Adjoint method is used to solve the sensitivity of the objective function , including according to the formula Solve the overall stiffness matrix K, the overall mass matrix M and the overall damping matrix C for the initial design variables The derivative of; and according to the formula Solve for P; where, let , is regarded as an adjoint vector, and P is an adjoint vector The transpose of Accumulate and get the derivative of the power spectrum density with respect to the design variable, according to the formula get ;

[0012] The moving asymptote method is used to solve the optimization problem, and the stopping criterion is set as follows: when the maximum difference of the design variables between two adjacent iterative steps is less than a preset value or the iterative step is greater than a preset value, the optimization process is terminated; if the stopping criterion is not met, the initial design variables are updated and the solution is returned to step 2 to continue.

[0013] In some embodiments, solving the modal characteristics of the marine engineering structure using a continuous iterative method based on analysis and design includes:

[0014] For the first iterative calculation, a random generation method is used to realize the input of the initial mode;

[0015] The overall stiffness matrix K and the overall mass matrix M are used to solve the formula , where the subscript k indicates the kth iteration, and k-1 indicates the k-1th iteration. represents the approximate mode of iteration to the kth time, Iterate to the k-1th modal matrix; through the required according to Update the q-order spatial stiffness and mass matrices, where and are the stiffness and mass matrices in the k-th q-order space respectively;

[0016] Solve the q-order generalized eigenvalue formula ;in and denote the eigenvalue and eigenvector of order q respectively;

[0017] Substitute the result of solving the characteristic equation into Update intrinsic modes, Represents the natural mode of iteration to the kth time;

[0018] Normalize the updated intrinsic modes and use the formula Update the approximate natural frequencies, represents the kth natural frequency.

[0019] In some embodiments, the marine engineering structure includes a jacket foundation structure of an offshore wind turbine, and the target model of the jacket foundation structure is a four-wing jacket structure; wherein the four-wing jacket structure is composed of four vertical rods and an X-shaped cross rod, and the X-shaped cross rod is connected between the four vertical rods.

[0020] In some embodiments, the optimization of the jacket foundation structure includes taking the root mean square of the horizontal displacement of the top center position of the jacket foundation structure as the objective function and the volume fraction as the constraint function; in the topology optimization process, assuming that the jacket structure is strictly fixed to the ground; applying the wave load equivalently in height, and simplifying the wind turbine on the top of the jacket structure to a point.

[0021] In some embodiments, the target model is set so that the overall size of the marine engineering structure remains unchanged, the four vertical rods are non-design domains, the other parts are design domains, and the X-shaped cross rods are the optimal area of ​​the plane continuum.

[0022] In some embodiments, as for boundary conditions, a fixed constraint is imposed at the bottom of the design domain for topology optimization, and a symmetry constraint is imposed in the middle of the design domain for topology optimization, thereby achieving simplification of the target model.

[0023] In some embodiments, the marine engineering structure includes a gangway structure, which includes a fixed part and a retractable part, and the load is loaded on the support of the retractable part and the fixed platform in the form of relative motion.

[0024] In some embodiments, the constraint function is a volume constraint; and a stationary white noise generated by the vertical motion of the fixed part and the fixed platform is loaded at the right end as a random excitation.

[0025] The beneficial effect of the present invention is that this method can solve the eigenvalues ​​and eigenvectors only once in a topology optimization iteration. Compared with the traditional method of solving the accurate eigenvalues ​​and eigenvectors of the structure multiple times in a single iteration, it greatly reduces the amount of calculation and improves computational efficiency. The present invention is suitable for the topology optimization design of structural dynamics under random loads. Optimization examples show that the proposed virtual excitation method based on analysis and design can obtain almost the same optimized design as the traditional double-loop virtual excitation method based on significant computational efficiency advantages. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 The figure is a simplified diagram of the optimization design process of marine engineering structures according to the present application.

[0027] Figure 2 This is the overall implementation flow chart of SIAD.

[0028] Figure 3 2D schematic diagram of the cantilever beam.

[0029] Figure 4 Schematic diagram of topology optimization results at different frequencies according to the present application, wherein part (a) is a schematic diagram of the topology optimization results at a frequency interval of [0~100] Hz, and part (b) is a schematic diagram of the topology optimization results at a frequency interval of [0~400] Hz.

[0030] Figure 5 : Iterative process diagram of different frequency target functions according to the present application, wherein part (a) is the iterative process diagram under the frequency interval of [0~100] Hz, and part (b) is the iterative process diagram under the frequency interval of [0~400] Hz.

[0031] Figure 6 Schematic diagram comparing topology optimization time for different degrees of freedom.

[0032] Figure 7 Schematic diagram of the support structure of the jacket structure, where (a) is the offshore wind turbine foundation support structure model; (b) is the jacket foundation elevation diagram and height dimensions.

[0033] Figure 8 Schematic diagram of the model and boundary conditions of the topology optimization method according to the present application.

[0034] Figure 9 FIG. 4 is a frequency iteration history diagram of the topology optimization according to the present application.

[0035] Figure 10 The optimized structure diagram of the jacket, where (a) is the topology optimized jacket structure, and (b) is the reconstructed topology optimized structure diagram.

[0036] Figure 11 It is the contour map of node displacement, where (a) is the reconstructed structural model and (b) is the original structural model.

[0037] Figure 12 The figure is a structural diagram of an existing gangway ladder, wherein part (a) is a cross section of the gangway ladder structure, and part (b) is a structural diagram of the retractable part of the gangway ladder.

[0038] Figure 13 Schematic diagram of the design domain and load conditions of the telescopic gangway cross section according to an embodiment of the present application.

[0039] Figure 14 Schematic diagram of the optimization results of the cross-section of a telescopic gangway ladder according to an embodiment of the present application, wherein part (a) is the topology optimization result of the telescopic gangway ladder section, and part (b) is the reconstructed topology optimization structure diagram.

[0040] Figure 15 3 is a node displacement contour diagram according to an embodiment of the present application, wherein part (a) is the original structural model and part (b) is the reconstructed structural model. DETAILED DESCRIPTION

[0041] This application proposes a topology optimization method for marine engineering structures under continuous iterative random excitation based on analysis and design. It is an efficient structural dynamic topology optimization method for complex random excitation working conditions. The root mean square (RMS) of the dynamic response displacement at specified key positions under complex random excitation is taken as the objective function.

[0042] To overcome the low computational efficiency of random excitation, this application employs the virtual excitation method (PEM). To address the high computational effort required to solve the natural frequencies and natural modes in each iteration of dynamic topology optimization, this application introduces an analysis and design sequence iteration to reduce computational time. The MMA gradient algorithm is used as the optimization algorithm. Given the high computational effort required to solve the objective function sensitivity using direct methods, this application employs the adjoint method (AMS) to solve the objective function sensitivity.

[0043] The proposed method is compared with the calculation example of the traditional double-loop random excitation topology optimization method. In the process of obtaining almost the same optimization design results, the PEM method based on analysis and design proposed in this application has a significant computational efficiency advantage over the traditional method. After the design optimization of two marine structures under complex working conditions using the method of this application, the offshore jacket foundation structure has obvious advantages over the original structure in terms of quality and objective function, while the offshore gangway structure not only reduces the distribution of rods but also optimizes the objective function while ensuring the same quality compared to the original structure. The overall optimization design has achieved significant improvements in dynamic performance compared to the traditional empirical design, which shows that this method has great potential in the design of marine engineering structures under complex environments.

[0044] The method of the present invention is described in detail below with reference to the accompanying drawings.

[0045] 1. Finite element analysis of dynamic systems under random excitation

[0046] During normal use, marine engineering structures are subject to random excitations from wind and waves. The original structure responds strongly under random excitations and has poor safety. In order to solve problems such as large responses, a topology optimization method is used to redesign the original structure. This application proposes a random excitation analysis and design sequence iteration method to perform topology optimization on the original structure design domain. The optimized structure is reconstructed and simulated to improve safety performance. The proposed conceptual design is detailed in the following. Figure 1 In order to better optimize the design of marine engineering structures, a dynamic structure finite element model will be established first.

[0047] 1.1. Finite element analysis of dynamic system.

[0048] For N-DOFs marine engineering structures, the finite element method is used to express its dynamic equation as follows:

[0049]

[0050] Where K, C, and M are the overall stiffness matrix, damping matrix, and overall mass matrix of the structure, respectively, and the dimensions are F(t) is an N-dimensional vector. C is the damping matrix and the classical damping matrix is ​​used. When solving the structural response, in order to reduce the calculation time, the modal displacement method (MDM) is used to approximate the exact response value with a small number of modes. Displacement response Expressed as:

[0051]

[0052] is composed of the first q-order modal vectors of the structure Modal matrix, z is composed of the amplitude of the first q-order modes of the structure Amplitude matrix, is the nth modal amplitude of the structure. In the case of classical damping, equation (1) can be decoupled into q uncoupled linear damped motion equations, and ensure that

[0053]

[0054] in, is composed of the square of the circular frequency of the q-order mode The diagonal matrix of is the damping ratio corresponding to the q-order mode. The n-th linear damped motion equation can be obtained by transformation:

[0055]

[0056] The modal amplitude in the time domain corresponding to the nth motion equation can be obtained through Duhamel integration:

[0057]

[0058] is the unit impulse response function associated with a single degree of freedom (DOF) system with the equation of motion:

[0059]

[0060] Through equations (2), (5) and (6), the displacement response expression in the time domain can be obtained, and the displacement response expression in the frequency domain can be obtained after Fourier transform operation:

[0061]

[0062] is the frequency domain transfer function between external excitation and structural response, .

[0063] In the above formula, is the modal vector under the nth mode, for The transposed matrix of is plural, Expressed as the unit transformation matrix, Expressed as the first vectors, Expressed as the frequency domain transfer function under the nth mode, is the complex exponential form of pseudo-harmonic excitation, is the damping ratio of the nth mode, is the circular frequency of the nth mode.

[0064] 1.2 PEM of dynamic systems under random excitation.

[0065] For the dynamic response of N-DOFs structure under single-point stationary random excitation, the dynamic equation can be written as:

[0066]

[0067] Where f(t) is a D-dimensional fixed random vector with non-zero values, and its power spectral density (PSD) for dimensional matrix. R is the unit transformation matrix, size is . PEM problem of single-point random excitation problem: When a linear elastic structure is subjected to a zero-mean steady-state random excitation, if and are two arbitrary response excitations generated by random excitation, then

[0068]

[0069]

[0070] Where * denotes conjugate, meaning the corresponding pseudo-harmonic response can be used and The auto-power spectral density (PSD) function of the corresponding random response can be obtained. Similarly, the pseudo-harmonic response of the two arbitrary responses can be obtained by and Find the cross power spectral density (CSD) function of the random response if the pseudo response is two arbitrary harmonic response vectors.

[0071]

[0072] Through equations (7) and (9), the pseudo response of the N-DOFs structure under single-point stationary random excitation can be obtained:

[0073]

[0074] in is the nth order frequency response function. According to PEM, the PSD matrix of X will be:

[0075]

[0076] The PSD matrix expression is completely equivalent to the quadratic combination (CQC) formula. Since the CQC formula involves all the coupling terms of the parametric vibration modes, for complex structures, the MDM method is used to reduce the order using q modes, and the computational complexity is also large. In order to reduce the computational complexity, the cross terms of equation (13) are usually ignored, and the square sum (SRSS) formula can be obtained:

[0077]

[0078] Although the SRSS method has improved computational efficiency compared to the traditional CQC method, it is only used for homogeneous material structures where all the parametric frequencies are sparsely distributed and the damping ratios of each modal order are very small. For general structures, the parametric frequencies are not sparsely distributed and the SRSS method cannot be used. Therefore, the SRSS method is relatively limited in use. Compared with the traditional CQC method and the SRSS method, the advantage of PEM lies in vector product. The traditional CQC method mainly performs The SRSS method only calculates the q-vector product, but its requirements are relatively stringent. The PEM method only calculates one vector product and guarantees the same accuracy as the traditional CQC method. Therefore, the PEM method is also called the fast CQC method.

[0079] For the dynamic response problem of N-DOFs structure under multi-point stationary random excitation, that is, considering the external stationary random excitation PSD as matrix .if is a Hermitian matrix, it can be decomposed into:

[0080]

[0081] in, yes rank, is a D-dimensional vector that can be obtained by Cholesky decomposition. The corresponding multi-point stationary external random excitation f(t) can be expressed as: , then the motion equation of multi-point random excitation can be expressed as:

[0082]

[0083] in, for By converting the unit vector, the pseudo-harmonic response expression can be obtained as:

[0084]

[0085] As a result, the corresponding PSD matrix of multi-point stationary random excitation will be:

[0086]

[0087] 2. Topology Optimization

[0088] 2.1 Optimization Model

[0089] According to the dynamic model under random excitation obtained by the PEM method, the RMS of the random displacement response of the rth N-DOFs is regarded as the objective function, and its topology optimization formula can be expressed as:

[0090]

[0091] in: is the frequency range of the external excitation; is the design variable of the e-th unit; is the volume of the e-th unit. The constraint condition set by the optimization formula is that the total volume must be less than the set volume requirement. The RMS of the objective function Expressed as: and The root mean square response of the structure under external excitation within the range is expressed by the power spectral density (PSD) It is obtained by integration and is used to measure the vibration amplitude of the structure under random excitation.

[0092] For the convenience of solution, the PSD of displacement can be expressed as:

[0093]

[0094] and

[0095]

[0096] in is a column vector with all degrees of freedom 0 except r, which is 1. In the multi-excitation problem, PSD is expressed as:

[0097]

[0098] and

[0099]

[0100] As mentioned above, the SIMP (Solid Isotropic Material Penalty Model) method model may encounter problems such as local modes when solving dynamic problems. In this application, the RAMP (Rational Material and Penalty Model) interpolation model is used for discretization, formula (24).

[0101]

[0102]

[0103] in, For unit Young's modulus, For unit density. and is the real Young's modulus and density of the material, as well as is the Young's modulus and density of the empty material, is the penalty factor.

[0104] When solving this objective function, even if the PEM method is used to solve the response to random excitation, it takes a long time, and the dynamic topology optimization itself requires multiple iterations to achieve a good distribution of materials. How to further reduce the optimization time of random excitation problems is a key issue.

[0105] 2.2 Analysis of random responses and design sequence iteration

[0106] When solving the equation of motion (1), it is unavoidable to solve the characteristic equation. Moreover, in solving dynamic problems, most of the time is consumed in solving the characteristic equation. In previous studies, general dynamic solution methods were mostly used, such as the inverse iteration method, the Lanczos method, and the subspace iteration method.

[0107]

[0108] This type of method improves computational efficiency by simply changing the solution of the characteristic equation of motion. In topology optimization, whether using the subspace iteration method or other methods, the characteristic equation must be solved multiple times in one optimization iteration. Although the computational time is reduced compared to directly solving formula (25), the time required is relatively long. Kang et al. proposed a method that is different from the traditional dynamic solution idea, namely, the Sequential Iterative Analysis and Design (SIAD) method. The specific idea is as follows: In one topology optimization iteration, a set of approximate natural frequencies and modes is used instead of the exact characteristic pairs. As the optimization progresses, the natural frequencies and modes will gradually approach the exact characteristic pairs, thereby achieving simultaneous optimization convergence of the objective function, natural frequencies, and modes.

[0109] In the solution, the SIAD method randomly selects an approximate eigenvector matrix , where q is the order of the selected subspace. In general, Among them, h is the order used in the calculation. Both q and h are much smaller than the structural freedom N. When the optimization iteration reaches k times, the modal matrix The solution can be performed by following the steps below:

[0110]

[0111] In the formula , is the overall stiffness and mass matrix at k iterations, is the modal matrix obtained for the k-1th time.

[0112]

[0113] in , is the eigenvalue of the subspace problem, is the eigenmode of the subspace problem.

[0114]

[0115]

[0116] Equation (27) is based on a q-dimensional subspace, significantly reducing the difficulty of solving the original equation. The approximate natural frequency is obtained using the Rayleigh quotient method in Equation (29). The design variables can be updated using a gradient-based optimization algorithm to proceed to the next optimization iteration. The SIAD method significantly reduces solution time compared to traditional dynamic topology optimization methods. Incorporating this method under random loads can further improve computational efficiency.

[0117] 2.3 Sensitivity Analysis

[0118] This application uses a gradient mathematical programming algorithm as the optimization solution. The sensitivity analysis of the objective function is to determine the derivative of the objective function with respect to a given design variable, which is necessary for the gradient mathematical programming algorithm to solve the optimization problem. Considering that the direct method requires a large amount of calculation to solve the sensitivity of the objective function, this application uses the adjoint method (ASM) to solve it. The sensitivity under multi-point steady excitation is almost the same as the sensitivity of the objective function under single-point excitation. For single-point steady excitation, the objective function can be obtained according to the optimization column (19):

[0119]

[0120] for It is expressed using formula (21):

[0121]

[0122] The derivative of formula (21) is expressed as:

[0123]

[0124] According to the equation of motion, we can get:

[0125]

[0126] Since the external load is independent of the design variables, we have , so the right side of formula (33) is 0.

[0127]

[0128] Formula (32) is expressed as:

[0129]

[0130] Among them , Consider it as a companion vector. Take the transpose of both sides of the companion vector equation and then multiply it by Then we have:

[0131]

[0132] Through the above derivation, we can get:

[0133]

[0134] For multi-point excitation, the sensitivity of the objective function can be expressed as:

[0135]

[0136] and

[0137]

[0138] 2.4 Optimize the process

[0139] Step 1: Given the initial design variable values ​​for each unit , solve the unit stiffness matrix and unit mass matrix according to the unit design variable value, and realize the generation of the overall stiffness matrix K, the overall mass matrix M and the overall damping matrix C according to the degree of freedom number and coordinate transformation.

[0140] Step 2: Use the SIAD method to solve the structural modal characteristics; for the first iterative calculation, the initial modal input can be realized by random generation or other methods, step S21; then the overall stiffness matrix K and M obtained in step 1 are used to solve formula (26), step S22; then the q-order generalized eigenvalue formula (27) is solved, step S23; the result of solving the characteristic equation is substituted into formula (28) to update the natural mode, step S24; the updated natural mode is normalized, and the approximate natural frequency is solved using formula (29), step S25.

[0141] Step 3: Solve for the PEM response value. Substitute the natural mode and natural frequency obtained in step 2 into formula (17) to solve for the pseudo-displacement response as well as , use formula (22) to get the displacement auto-power spectrum density value of the r-th degree of freedom point, and then use formula (19) to get the RMS of the objective function, so as to achieve the response solution of random excitation.

[0142] Step 4: Use the AMS method to solve the sensitivity of the objective function. According to formula (39), it is necessary to solve the overall stiffness matrix K, the overall mass matrix M and the overall damping matrix C for the design variables The derivative of . And solve P according to formula (36), and the obtained Accumulate and get the derivative of PSD with respect to design variables. According to formula (30), we can get .

[0143] Step 5: Use an iterative optimization method such as the moving asymptote method (MMA) to solve the optimization problem, and set the stopping criterion as: when the maximum difference of the design variable between two adjacent iteration steps is less than a preset value, such as the specified value , or when the iteration step is greater than the preset value, such as a certain value, the optimization process will terminate; if the set stopping criteria are not met, after updating the design variables, return to step 2 and continue solving until the conditions are met. For the specific topology optimization implementation flow chart, see Figure 2 .

[0144] 3 Calculation examples

[0145] 3.1 Topology optimization of 2D cantilever beam structure

[0146] In order to verify the feasibility of the analysis and design sequence iterative topology optimization method under random excitation, a 2D cantilever beam structure is used as an example to demonstrate the effectiveness and computational efficiency of this method.

[0147] The 2D cantilever beam structure is a rectangular design domain based on a unit size of 1m and a thickness of 0.001m, as shown in Figure 3 As shown, the left side is a fixed constraint, and a PSD value is added to the middle node on the right. The Young's modulus, Poisson's ratio and density are 78GPa, 0.3 and .

[0148] The feasibility of this method will be demonstrated from three aspects: change in external excitation frequency; change in the number of modes, and change in the number of units.

[0149] The following examples are all run in the MATLAB environment.

[0150] 3.1.1 Topology optimization of 2D cantilever beam structure under changing external excitation frequency.

[0151] This section selects two external excitation frequency ranges to discuss the impact of the SIAD optimization method under random excitation as the external excitation frequency range changes. Figure 3 The structure is set with the minimum root mean square displacement response of the loading point as the objective function and the ratio of the solid material volume to the design domain volume being less than or equal to 0.5 as the constraint function. Set to 0.5, with a unit number of 140 70 is used as the optimized structure, and the penalty factor of RAMP interpolation model is 8. The number of modes selected by MDM is 20. The structure is excited using two frequency intervals [0~100] Hz and [0~400] Hz. The results are shown in Figure 4 The optimization process diagram is shown in (a) and (b). Figure 5 (a) and (b) of .

[0152] Through the optimization results, it is found that as the frequency range becomes larger, the topology optimization results will transform from thickened components to complex bending component design. Figure 5 The root mean square displacement of the optimized structure decreased continuously during the iterations. For the frequency interval [0-100] Hz, the objective function was optimized from 0.7575 m to 0.1088 m after 324 iterations. Throughout the optimization iterations, the restraint and loading positions were optimized first. As the iterations progressed, the bending moment generated by the applied vertical excitation increased, and an "X"-shaped crossbar structure, which resisted the moment, began to be optimized, generating the main structure. Subsequently, as the number of iterations increased, the structure began to optimize the robust members, while the moment-resisting structure was simplified to generate thinner members. For the frequency interval [0-400] Hz, the objective function was optimized from 9.8098 m to 0.2223 m after 314 iterations. Because the fundamental frequency of the structure fell within the external excitation frequency range, the restraint and loading positions were not optimized initially at the beginning of the optimization, as they were for the [0-100] Hz optimization. The results only become apparent after several iterations. Due to the wide range of external excitation, the structure focuses more on local structural optimization. As the frequency interval increases, local vibrations are increased, and to reduce the RMS displacement, local stiffness must be increased. This results in a more "X"-shaped crossbar structure than at lower frequency intervals. Furthermore, at low-frequency intervals, the structure tends to thicken within the existing structure during the iteration process to reduce the objective function. However, at high-frequency intervals, the impact of greater local flexibility on the objective function must be considered, leading to a more complex structure to reduce the objective function.

[0153] 3.1.2 Topology optimization of 2D cantilever beam structure with varying modal number

[0154] This section selects the third-order mode number to compare the computational efficiency of the SIAD optimization method under random excitation and the subspace iteration method under random excitation. For random excitation topology optimization problems, the computational time is mainly affected by the number of modes involved and the number of degrees of freedom. When MDM approximates the structural objective function, it ignores the influence of high-order modes. When the number of selected modes q continues to increase, the accuracy of the objective function will be improved. However, as the number of modes increases, the difficulty of solving the characteristic equation and the amount of calculation also increase exponentially. Therefore, the time used by the SIAD method under the change of the number of modes is compared with the time used by the subspace iteration method. Based on Figure 3 The total number of units is 140 70, the penalty factor of the RAMP model , the external excitation is controlled in the range of [0~100] Hz, and the subspace iteration method is used as a control. The error Take Three modes, q, with modal numbers of 20, 30, and 40, were selected for comparison, and the results are shown in Table 1. Table 1 shows that as the modal number q increases, the time taken to perform topology optimization using the subspace iteration method is significantly higher than that of the SIAD method under single-degree-of-freedom excitation. Although the subspace iteration method obtains more accurate eigenvalues ​​and natural modes in each optimization iteration, the target values ​​obtained by the SIAD method and the subspace iteration method differ by less than 1%. As the number of modes changes, the SIAD method remains almost unchanged, while the number of computational events for the subspace iteration method increases. Furthermore, the computational efficiency of the SIAD method can be improved by up to 60.63% compared to the subspace iteration method.

[0155] Table 1 Comparison of topology optimization time for different numbers of modes

[0156]

[0157] 3.1.3 Topology Optimization of 2D Cantilever Beam with Varying Number of Degrees of Freedom

[0158] This section selects three numbers of degrees of freedom to compare the computational efficiency of the SIAD optimization method under random excitation and the subspace iteration method under random excitation. Considering the time-consuming problem of the constantly changing number of degrees of freedom, when the structure is analyzed in detail, the number of units it is divided into is quite large, which will lead to situations such as tens of thousands of degrees of freedom, making the computational efficiency extremely low. When performing dynamic optimization, as the number of units increases, the DOFs become larger, and the time required for the characteristic equation of the original structure usually cannot be ignored. Even if the MDM method is used for simplification, the computational time required is still large, so by changing the number of units, the number of DOFs is divided into three parts, namely: number of degrees of freedom -1: 20022; number of degrees of freedom -2: 26082; number of degrees of freedom -3: 32942. The SIAD method is compared with the subspace iteration method, and the results are shown in Table 2. As the number of degrees of freedom continues to increase, the time used by both the SIAD method and the subspace iteration method continues to increase. However, through Figure 6 It can be seen that the SIAD method takes less time than the subspace iteration method. Furthermore, the SIAD method achieves a maximum efficiency improvement of 66.19% compared to the subspace iteration method. When performing topology optimization on large structures, increasing the number of elements often increases the number of modes to improve computational accuracy. Comparing the number of modes and the number of DOFs shows that SIAD can significantly improve computational efficiency when performing topology optimization. Furthermore, the variation in the external (external load) excitation frequency demonstrates the SIAD method's wide applicability to random excitations.

[0159] Table 2 Comparison of topology optimization time with different numbers of degrees of freedom

[0160]

[0161] The feasibility and high computational efficiency of the SIAD method under random excitation are demonstrated by considering the external excitation frequency range, number of modes, and number of degrees of freedom. The method can effectively generate topological configurations across the external excitation frequency range from low to high frequencies. Furthermore, as the number of modes and degrees of freedom increases, the SIAD method demonstrates significant computational efficiency advantages over traditional optimization methods. The feasibility and advantages of this method are demonstrated using a 2D cantilever beam structure as an example, providing strong support for the optimization of practical offshore engineering structures.

[0162] 3.2 Offshore Jacket Foundation Topology Optimization

[0163] 3.2.1 Jacket foundation structure

[0164] This section will conduct topological optimization of the jacket foundation structure (abbreviated as jacket structure) of offshore wind turbines (OWT) in marine engineering structures. As a common foundation type for offshore wind turbines, the jacket structure has the advantages of less material consumption and strong load bearing capacity. It is widely used in near and shallow waters. Figure 7 As shown in part (a) of the figure. The target model is a four-wing jacket structure, which is used in a water depth of 45m. The overall structure of the offshore wind turbine mainly consists of a wind turbine, a concrete transition part, a support structure and a pile foundation. The upper wind turbine, which mainly includes a tower, blades and a generator set, was simplified in the initial stage of finite element modeling. The generating capacity of the generator set is 5MW, and the total mass of the wind turbine is 350,000 kg. The height value is based on the sea level of 0m as a reference. Figure 7 As shown in part (b), the jacket structure is mainly composed of four vertical rods, and four layers of X-shaped cross rods are used to connect every two vertical rods to the four vertical rods. The dimensions of each rod are shown in Table 3, which are 280 meters long and 140 meters high.

[0165] Jacket structure load calculation. Jacket structures are primarily subject to wind loads, wave loads, and surge loads. There are two types of random loads acting on the jacket structure: one is the random load transmitted from the wind load on the wind turbine tower to the top of the foundation structure; the other is the random load acting directly on the supporting structure from the environment, with the wave load being considered the primary random load. The Von Karman wind spectrum is used to simulate random wind loads, and its power spectrum density formula is:

[0166]

[0167] in: is the power spectral density of wind speed, n is the frequency, is the standard deviation of wind speed, is the longitudinal turbulence integral scale, and U is the average wind speed.

[0168] The time domain wind load formula is:

[0169]

[0170] in: is the wind load on the structure, is the air density, A is the windward area, and in a vertical axis fan the cross-sectional area of ​​the tower is taken; Time domain wind speed.

[0171] The Pierson-Moskowitz spectrum is used to simulate wave loads. The PM spectrum is widely used in marine engineering, ship design, and offshore structure load calculations due to its clear parameters and efficient calculation. Its power spectrum density formula is:

[0172]

[0173] in: is the power spectral density of the Pierson-Moskowitz spectrum, is a dimensionless constant, usually taken as 0.0081; g is the acceleration due to gravity; is the peak frequency; is the frequency;

[0174] Wave force formula: Wave loads are mainly generated by the dynamic pressure and inertial force generated by wave velocity and wave acceleration on the structure:

[0175]

[0176] in: is the wave load on the structure; is the water density; A is the cross-sectional area of ​​the structure in water; are the wave velocity, the structure's motion speed, and the wave acceleration; are the drag coefficient and the mass coefficient.

[0177] Table 3. Member properties of the jacket structure

[0178]

[0179] 3.2.2 Topology optimization and optimized reconstruction structure

[0180] Based on the topology optimization model in Section 3, that is, the model represented by formula (19), the substructure under dynamic random loads is optimized, and the structure is reconstructed according to the topology optimization structure and analyzed with the existing model. The horizontal displacement RMS (root mean square) of the top center position of the jacket foundation structure is used as the objective function, and the volume fraction is used as the constraint. In the topology optimization process, it is assumed that the jacket structure is strictly fixed to the ground. In order to simplify the optimization model and improve the calculation efficiency, the wave load is applied equivalently in height, and the upper wind turbine is simplified to a point, such as Figure 8As shown, there are two random excitations. Considering that wind and wave loads have no coupling effect on the jacket structure, the off-diagonal elements of the external load random excitation matrix are zero. The allowable optimization region is determined based on the existing structure. Considering the characteristics of the jacket structure, the overall structural dimensions remain unchanged, the four vertical members are the non-design domain, and the X-shaped crossbars are the optimal region of the planar continuum. For boundary conditions, fixed constraints are applied at the bottom of the topology optimization design domain, and symmetry constraints are applied in the middle of the topology optimization design domain to simplify the target model.

[0181] Using the SIAD topology optimization method under random excitation, a new conceptual design that meets the design requirements was obtained by analyzing the frequency iteration history diagram. According to the design criteria for offshore wind power infrastructure, its first-order natural frequency is usually between 1P (rotor speed) and 3P (blade passing). The speed range of the OWT is 7.5~14 r / min, and the circular frequency ranges corresponding to 1P and 3P are 0.785~1.464 rad / s and 2.356~4.398 rad / s, respectively. P is the rotor speed of the horizontal axis three-blade wind turbine. Considering 10% redundancy to prevent resonance, the reasonable fundamental frequency range is between 1.610 and 2.121 rad / s. The Young's modulus, Poisson's ratio, and density of the topology optimization design domain are 210GPa, 0.3, and 7850kg / m, respectively. 3 , the penalty factor of the RAMP model , topology optimization iteration diagram of jacket structure under SIAD random excitation, see Figure 9 During the optimization process, the circular frequency first increased to a relatively high level. Then, as the optimization progressed, the horizontal forces caused by wave loads and wind loads played a dominant role. The upper part was the first to appear in the diagonal rod structure, the vertical rods of the lower structure became thicker, and the frequency began to decrease. Due to the continuous iteration of the optimization, the bending moment load transmitted by the tower top played a dominant role. The lowermost structure was the first to appear in the diagonal rod support. After iteration, the middle diagonal rod support structure also appeared, and the upper diagonal rod began to become thicker. Finally, after 286 iterations, the circular frequency of the optimized structure was 1.857rad / s, which met the design requirements. The optimization results are shown in the figure. Figure 10 As shown in part (a).

[0182] The final design obtained based on the topology optimization in 3.2.3 cannot be used directly for manufacturing. The topology optimization results need to be interpreted and reconstructed, and compared and analyzed with the existing structure. The topology optimization results are reconstructed based on the symmetry of the four sides of the jacket. Figure 10 The optimized materials of the middle and lower vertical members shown in part (a) are more, and they are thicker than the upper vertical members. Therefore, the lower vertical members of the optimized structure are appropriately thickened. Figure 7, some seemingly redundant brackets were removed, and the size of the X-shaped crossbar support was appropriately adjusted, resulting in a 46.46% reduction in the mass of the reconstructed structure (248.272t) compared to the original structure mass (463.704t). Figure 10 As shown in part (b), the geometric parameters are shown in Table 4. The displacement root mean square is recalculated according to the random load used in 3.2.3 and compared with the existing structural model. Figure 11 ,like Figure 11 As shown in the figure, under the same load, the upper displacement of both the existing structural model and the reconstructed structural model is larger, which is consistent with the actual situation. Figure 11 In the existing structural model shown in part (b), most of the member displacement RMS is above 0.2874m, and the overall member displacement RMS varies greatly. Figure 11 In the reconstructed structural model of part (a), the displacement of the lower member becomes smaller, the RMS displacement of the entire structure changes more evenly, and the high displacement RMS area is also smaller than the existing structural model. Figure 11 The maximum RMS displacement of the middle (a) part is 0.3374m, while Figure 11 The maximum RMS displacement of part (b) is 0.4311m, which is a decrease of 21.7% compared with the previous figure.

[0183] Table 4. Member properties of the jacket structure

[0184]

[0185] 3.3 Gangway topology optimization

[0186] 3.3.1 Gangway structure model

[0187] In this section, we will introduce the existing gangway structure, perform topological optimization on the retractable part of the gangway structure, reconstruct the gangway structure and analyze it. Transferring from a ship to a fixed offshore platform is difficult. The wave force on the ship causes the ship to move continuously, while the offshore platform is almost stationary. The relative movement between the ship and the structure makes it difficult to transport personnel and cargo. A gangway is a device that can safely transport personnel and cargo from a floating ship to another offshore structure. The gangway structure can be roughly divided into two parts: a fixed part and a retractable part, see Figure 12 (a) The gangway structure is entirely steel, with one end attached to the offshore platform frame and bolted to the platform frame. The upper portion is connected to the platform frame via a hydraulic device, which adjusts the gangway structure's rotation perpendicular to the water surface. A motor is installed below the fixed portion, connected to a rope, to achieve dynamic linear motion of the retractable portion.

[0188] The key dimensions, working conditions and load distribution of the gangway structure are described. The existing gangway structure dimensions are shown in Table 5. The component dimensions are divided into 5 parts. The cross-sectional dimension parameters of each part are consistent. Figure 12 (b). The overall length, width and height of the retractable part are respectively Under normal operating conditions, the gangway structure is in working mode. This means that the gangway structure is capable of transferring personnel from one offshore structure to another. During the retraction and deployment phase, i.e. when the gangway is in the raised state, no personnel or cargo is transported on the gangway structure. The main load on the gangway structure consists of its own weight acting on the gangway structure. For emergency disconnection, the main load consists of the gangway structure's own weight load and the live load caused by personnel and cargo, which should be applied to the connection between the retractable section and the fixed platform.

[0189] Table 5 Dimensions of the retractable part

[0190]

[0191] 3.3.2 Topology optimization and optimized reconstruction structure

[0192] In this example, the gangway is considered to be in a normal state, that is, the gangway structure is in a normal working mode. At this time, one end of the gangway structure is connected to the hull and the other end is in contact with the fixed structure at sea. The retractable part in this situation is topologically optimized, and the topological optimization model is interpreted and reconstructed, and compared with the existing gangway structure. The entire design space is surrounded by four rectangles. Structure, such as Figure 13 As shown in the figure, 1, 2, and 3 are translational motions along the X, Y, and Z axes, while 4, 5, and 6 are rotational motions along the X, Y, and Z axes. The dark part is the non-design domain. Since the gangway structure is symmetrical, the design domain is designed symmetrically to improve the calculation efficiency and accuracy. When the retractable part is fully extended, its constraints are as follows: Figure 13 As shown, the material is steel, and the Young's modulus, Poisson's ratio and density of the material are 210GPa, 0.3 and 7850kg / m respectively. 3 , the penalty factor of the RAMP model Under normal circumstances, the acceleration of the gangway structure perpendicular to the water surface is 11.35m / s 2Since the gangway structure is connected to the fixed platform, the gangway structure and the fixed platform undergo relative motion. The mass of the fixed part and the platform is approximately 410 kg. Under the action of vertical acceleration, the mass of the fixed part and the fixed platform will generate an impact load on the retractable part. In order to facilitate the solution, the load is loaded on the support of the retractable part and the fixed platform in the form of relative motion. The constraint function is a volume constraint, and the volume fraction is set to 0.5. The right end is loaded with a stationary white noise random excitation generated by the vertical motion of the fixed part and the fixed platform, and the load frequency range is [0~1] Hz. Since the maximum value of the PSD generated by the Pierson-Moskowitz spectrum is between 0 and 1 Hz, when the frequency is greater than 1 Hz, the waves are small and it is considered that the gangway remains stationary.

[0193] After optimization iterations, the topology optimization configuration is obtained. It can be found that the top view of the structural configuration has material distribution, and the front view and bottom view have reduced the material distribution of the existing structure. The results obtained through topology optimization are as follows: Figure 14 As shown in part (a) of the figure, the optimized structure shows that the vertical members alone cannot support the structure on the right side of the front view under random excitation. The bending moment generated by the random excitation vertical force has a significant impact, resulting in a higher distribution of material in the upper left diagonal section of the front view, as well as increased material at the lower and upper constraint points. This excess material in the lower section results in less material in the lower diagonal member connecting the upper constraint in the middle section. Near the load location, the loading is within the non-design domain, where the non-design domains touch each other, resulting in a lack of material distribution in the front view section near the load application. In the middle of the front view, a diagonal member with thicker material distribution connects the top and bottom views, improving overall stability. This thicker material distribution also indicates that simply connecting the upper and lower sections at the load application point is insufficient. The material distribution generated in the front view also indicates that under this load condition, more diagonal braces and vertical members are not actually necessary. In the top view, material is added from the constraint section toward the load application point during optimization, connecting the middle section to the diagonal member with thicker material distribution in the front view. Although there is an off-design domain between the restraint position and the center, the resulting material distribution does not rely on this off-design domain. Instead, it forms an "X"-shaped crossbar structure, indicating that the bending moment caused by the load is the primary influence. Compared to the original structure with no upper structure, the double "X" crossbar structure formed by topological optimization in the top view improves the overall structure's ability to withstand the moment caused by the load. In the bottom view, the crossbars are reduced compared to the original structure, and the material distribution is concentrated near the load location.

[0194] Since the topology optimization results cannot be used normally in actual situations, the topology optimization structure is interpreted and rebuilt based on the topology optimization results, and compared and analyzed with the existing structure. Figure 14 As shown in the middle (b), the dimensions of each member are shown in Table 6. Since the number of members in the optimized structure is reduced compared to the original model, the thickness of the members will be adjusted so that the mass of the retractable part after the transformation is 223.359t, which is 9.850% different from the original structure mass of 247.765t. The RMS is recalculated according to the random load used in the topology optimization and compared with the existing structural model. Figure 15 As shown in parts (a) and (b), under the same excitation conditions, Figure 15 The maximum RMS value of the original model shown in part (a) is 12.05 mm. Figure 15 The maximum RSMS value of the optimized model shown in the middle (b) is 8.47mm. Under the condition of the same mass, it is reduced by 29.417%. Analyzing the displacement cloud diagram, the original model has a greater vibration response than the optimized model. Obviously, the topology optimization makes the structural design more reasonable. And in the displacement cloud diagram, Figure 15 The displacement distribution of the structure shown in part (b) is compared with Figure 15 The structure shown in part (a) is more uniform and has no obvious local deformation. Figure 15 The structure shown in part (a) uses more rods, but they do not have much effect under this random excitation.

[0195] Table 6 Optimized structural dimensions of the retractable part

[0196]

[0197] It should be understood that the numerical values ​​and numerical ranges in the specific embodiments provided herein are not intended to be limiting and may be adaptively adjusted based on the structural characteristics of different marine engineering structures. Furthermore, other equivalent numerical values ​​and numerical ranges that achieve the same, similar, substantially the same, or substantially similar effects also fall within the scope of protection of the present invention. For example, the fundamental frequency range, penalty factor, number of iterations, volume fraction, PSD, etc. described above may also be within other ranges, such as larger or smaller.

[0198] In view of the fact that dynamic analysis and topology optimization are performed sequentially in topology optimization problems under random loads, the present invention updates the approximate natural modes in the analysis process, and then introduces the updated approximate vectors into the optimization process to update the design variables and solve the characteristic equations. The update mode of the approximate natural modes adopts a quasi-inverse iteration mode, which only requires solving a set of linear equations, and the total number of approximate vector updates is the same as the number of optimization iterations. This method can significantly reduce the amount of calculation and improve computational efficiency. It makes equivalent replacements for the objective function of the optimization problem described in the aforementioned embodiments, the optimized structure, or the material interpolation model therein, without causing the essence of the corresponding methods and solutions to deviate from the scope of the methods and solutions of the various embodiments of the present invention.

[0199] This application proposes a topology optimization method for rapid solution through iterative analysis and design sequences under random excitation, and applies it to the optimization of offshore jacket foundation structures and offshore gangway structures. Engineering applications often face complex working conditions, and incorporating random excitation into the structural design phase can significantly impact structural safety.

[0200] The traditional CQC method suffers from low computational efficiency in random excitations. While the SRSS method improves computational efficiency, its simplified form requirements prevent its widespread use. This application utilizes the PEM method, effectively addressing the low computational efficiency of the traditional CQC method and the limitations of the SRSS method.

[0201] In the field of dynamic topology optimization, facing local modal problems, this application uses the RAMP interpolation model to solve the precise inherent characteristics of the structure, which requires a long calculation time. In traditional dynamic topology optimization, each optimization usually requires solving the precise characteristic frequency and mode, usually requiring solving multiple linear equations, which is computationally inefficient. The SIAD method has advantages in dealing with problems such as the long time required to solve inherent characteristics. Since the MMA method requires solving the sensitivity of the objective function for optimization, this application uses the adjoint method to solve the sensitivity of the objective function to address the low efficiency of the direct method, thereby establishing a topology optimization model.

[0202] The above article elaborates on the application process of the method of the present application, and compares it with the traditional double-loop solution method under random excitation, highlighting its significant advantages in computational efficiency. This method has a wide range of applicability and its efficiency is greatly improved compared to traditional methods. Since marine engineering structures face many random excitation problems, this application uses this method to perform topology optimization on two marine engineering structures. Among them, after using this method on the offshore wind turbine jacket foundation structure, the mass of the optimized structure is reduced by 46.46% compared with the mass of the original structure, and the objective function is reduced by 21.7% in comparison. In the offshore corridor bridge structure, after using this method on the retractable part, while ensuring that the mass is almost equal, the objective function is reduced by 29.417% in comparison. These two engineering structure examples reflect the advantages of the topology optimization method of rapid solution of analysis and design sequence iteration under random excitation proposed in this application and the effect of this method in the topology optimization of marine engineering structures.

Claims

1. A topology optimization method for marine engineering structures under continuous iterative random excitation based on analysis and design, characterized by: Establishing a dynamic model of the marine engineering structure with N degrees of freedom, wherein the dynamic model includes a plurality of units; According to the dynamic model under the random excitation obtained by the virtual excitation method, the root mean square of the random displacement response of the rth degree of freedom in the N-dimensional degrees of freedom is regarded as the objective function, and the topology optimization formula of the dynamic model is obtained: ； in, is the frequency of external excitation scope; is the design variable of the e-th unit; is the volume of the e-th unit; the constraints set by the optimization formula are: the total volume must be less than the set volume requirement; the root mean square of the objective function Expressed as: and Under the external excitation within the range, the power spectrum density The integral is obtained to measure the vibration amplitude of the marine engineering structure under the random excitation; V is the established volume requirement; f( t ) is non-zero D dimensional fixed random vector, whose autopower spectrum density is dimensional matrix; R is the unit transformation matrix, size is ; Given the initial design variable values ​​for each of the units , solving the unit stiffness matrix and the unit mass matrix according to the initial design variable values, and generating the overall stiffness matrix K, the overall mass matrix M and the overall damping matrix C of the dynamic model according to the degree of freedom number and coordinate transformation; solving the modal characteristics of the dynamic model using a continuous iterative method based on analysis and design, wherein the modal characteristics include natural modes and natural frequencies; Substituting the obtained natural mode and natural frequency into the formula Among them, represents the modal vector under the nth mode, for The transposed matrix of represents the unit transformation vector, represents the kth vector obtained by Cholesky decomposition, represents the frequency domain transfer function under the nth mode, The complex exponential form of pseudo-harmonic excitation is used to solve the overall displacement response of the marine engineering structure under the kth pseudo-harmonic The displacement response of the rth degree of freedom of the marine engineering structure under the kth pseudo-harmonic , using the formula Get the displacement power spectrum density value of the rth degree of freedom , and then calculate the root mean square of the objective function to achieve the response solution to the random excitation; l is rank; Adjoint method is used to solve the sensitivity of the objective function , including according to the formula Solve the overall stiffness matrix K, the overall mass matrix M and the overall damping matrix C for the initial design variables The derivative of ; and according to the formula, Solve for P; where, let , is regarded as an adjoint vector, and P is an adjoint vector The transpose of Accumulate and get the derivative of the power spectrum density with respect to the design variable, according to the formula get ; The moving asymptote method is used to solve the optimization problem, and the stopping criterion is set as follows: when the maximum difference of the design variables between two adjacent iterative steps is less than a preset value or the iterative step is greater than a preset value, the optimization process is terminated; if the stopping criterion is not met, the initial design variables are updated and the solution is returned to step 2 to continue.

2. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 1, characterized in that: The method of solving the modal characteristics of the marine engineering structure using a continuous iterative method based on analysis and design includes: For the first iterative calculation, a random generation method is used to realize the input of the initial mode; The overall stiffness matrix K and the overall mass matrix M are used to solve the formula , where the subscript k indicates the kth iteration, and k-1 indicates the k-1th iteration. represents the approximate mode of iteration to the kth time, Represents the modal matrix of the iteration to the k-1th time; through the required according to Update the q-order spatial stiffness and mass matrices, where and are the stiffness and mass matrices in the k-th q-order space respectively; Solve the q-order generalized eigenvalue formula ;in and denote the eigenvalue and eigenvector of order q respectively; Substitute the result of solving the characteristic equation into Update intrinsic modes, Represents the natural mode of iteration to the kth time; Normalize the updated intrinsic modes and use the formula Update the approximate natural frequencies, represents the kth natural frequency.

3. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 1, characterized in that: The marine engineering structure includes a jacket foundation structure of an offshore wind turbine, wherein a target model of the jacket foundation structure is a four-wing jacket structure; wherein the four-wing jacket structure is composed of four vertical rods and an X-shaped cross rod, and the X-shaped cross rod is connected between the four vertical rods.

4. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 3 is characterized by: The optimization of the jacket foundation structure includes taking the root mean square of the horizontal displacement of the top center position of the jacket foundation structure as the objective function and the volume fraction as the constraint function; in the topology optimization process, it is assumed that the jacket structure is strictly fixed to the ground; the wave load is applied equivalently in height, and the wind turbine on the top of the jacket structure is simplified to a point.

5. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 3 is characterized by: The target model is set such that the overall size of the marine engineering structure remains unchanged, the four vertical rods are non-design domains, the other parts are design domains, and the X-shaped cross rods are the optimal area of ​​the plane continuum.

6. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 5, characterized in that: As for boundary conditions, a fixed constraint is imposed at the bottom of the design domain of topology optimization, and a symmetric constraint is imposed in the middle of the design domain of topology optimization, thereby achieving simplification of the target model.

7. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 1, characterized in that: The marine engineering structure includes a gangway structure, which includes a fixed part and a retractable part. The load is loaded on the support of the retractable part and the fixed platform in the form of relative motion.

8. The method for topology optimization of marine engineering structures under continuous iterative random excitation based on analysis and design according to claim 7, characterized in that: The constraint function is a volume constraint; and the stationary white noise generated by the vertical motion of the fixed part and the fixed platform is loaded at the right end as the random excitation.

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