Structural seismic response calculation method based on dynamic guidance support vector machine

Abstract

The application discloses a structural seismic response calculation method based on a dynamic guidance support vector machine, which comprises the following steps: a finite element model is established to extract structural dynamic characteristics; in combination with the structural dynamic characteristics, a single degree of freedom system motion equation is used as a constraint condition to optimize an objective function, and parameter optimization of a prediction model is realized; an LS-SVM algorithm is adopted to input structural characteristics and ground motion, and to output response quantities such as displacement and velocity, so as to construct a PDLS-SVM model; a modal decomposition method is used to decouple a multi-degree of freedom system, and a dynamic equation and a modal superposition principle are used to construct a seismic response prediction model; and based on a linear interpolation recursive formula, the performance of the seismic response prediction model of the structure is verified, and performance evaluation indexes are obtained; the application uses structural dynamic equations and physical laws as a basis, and considers structural dynamic characteristics to construct a data-independent seismic prediction model, so that the shortcomings of traditional methods and machine learning methods are overcome, and the calculation efficiency and accuracy are improved.

Description

TECHNICAL FIELD

[0001] The application relates to the field of structural engineering and anti-seismic engineering, and in particular to a structural anti-seismic response calculation method based on a dynamically guided support vector machine. BACKGROUND

[0002] With the development of urban disaster resistance and safety, the field of structural anti-seismic engineering is gradually expanding from the concern for single structures to the entire urban building group. Among them, quickly and accurately predicting the dynamic response of urban high-rise buildings under the action of earthquakes has become a key technical challenge in the disciplines of structural engineering and anti-seismic engineering. Structural dynamic response analysis provides a basis for determining the bearing capacity and dynamic characteristics of structures and improving the seismic design of structural performance.

[0003] Since the early 1970s, traditional numerical calculation methods have been widely used in engineering practice. Direct integration methods have obvious drawbacks due to theoretical assumptions, so these methods generally only have first-order or second-order accuracy, especially in the high-frequency stage, which is less accurate. In view of the defects of traditional numerical calculation methods, in 1994, Academician Zhong Wanxie of China proposed a fine time-history integration method. Due to its high calculation accuracy and efficiency, this new step-by-step integration method has attracted widespread attention in the academic community. This method fully utilizes the characteristics of the matrix exponential function that can be accurately calculated within the computer word length range to obtain a high-precision solution of the dynamic equation. However, for large structures in practical engineering, due to the large number of degrees of freedom, and the fact that the exponential matrix is generally full, the storage capacity and computational workload increase substantially, limiting the application of fine integration methods. In order to solve the problem of the large scale of the exponential matrix, Academician Zhong Wanxie further proposed a subdomain fine integration method, which not only has the advantages of unconditional stability and high precision of fine integration, but also has the benefit of small bandwidth. With the improvement of people's material living standards, people have more and higher requirements for the form and height of building structures. Large and complex structural systems can be seen everywhere, and the fine integration method is not enough to completely solve the problem of seismic response analysis of complex structures, and the current analysis method for such structures is generally theoretical analysis using the finite element method.

[0004] Many regions of the world, especially those located in seismic zones, have a great need for earthquake-resistant structural design. These structures must be able to protect people and reduce economic losses when earthquakes occur. Building structural design must meet a variety of needs, including durability, functionality, economy, and safety. Traditional numerical integration methods are often inefficient in dealing with high-dimensional, multi-constrained seismic design problems. In recent years, advanced technologies represented by machine learning algorithms such as neural networks, deep learning, and support vector machines have been successfully applied to building seismic response prediction, making up for the shortcomings of classical methods. These methods, through a data-driven learning process, do not rely on traditional approximation theory, and can improve computational efficiency while maintaining high prediction accuracy. For example, the Journal of Structural Engineering in March 2021, "Seismic Drift Demand Estimation for Steel Moment Frame Buildings: From Mechanics-Based to Data-Driven Models," established a framework for developing hybrid or data-driven models to estimate structural response under extreme events. The Earthquake Engineering & Structural Dynamics journal in March 2023, "Combination of physics‐based and data‐driven modeling for nonlinear structural seismic response prediction through deep residual learning," proposed a discrete physics-DNN hybrid integration time recorder to calculate the time evolution of nonlinear structural dynamic systems under seismic excitation. Machine learning methods adopt a data-driven modeling strategy, which eliminates prior assumptions about function form. Unlike modeling methods that rely on traditional approximation theory, machine learning techniques can learn and build models directly from data itself without pre-defining or assuming mathematical relationships between data.

[0005] While machine learning methods have demonstrated their powerful data processing and prediction capabilities in many fields, they also have some drawbacks and limitations in application. First, machine learning methods are highly dependent on data, which means that the performance of the model is directly affected by the quality and quantity of the data. If the data set has bias, noise or is incomplete, it may cause the model to learn incorrect information, which will affect the accuracy of the prediction. Second, machine learning models, especially deep learning models, are often criticized as "black box" models because their decision-making process lacks transparency and interpretability. This poses a challenge for fields that require high interpretability, such as medicine, financial security, etc., so it is difficult for decision-makers to fully understand the basis of the model's prediction. Third, the selection of hyperparameters has a significant impact on the performance of machine learning models. However, there is no general rule or method to guide the optimal selection of hyperparameters, which makes the tuning process of the model often require a lot of trial and error and experience, which not only consumes time but also may not achieve optimal performance due to lack of experience. Finally, the generalization ability of machine learning models is also a key problem. The model may perform well on the training set, but its performance may decline on new data that has not been seen before, which is a manifestation of overfitting. Therefore, although machine learning technology has great potential in data analysis and prediction modeling, its limitations in data dependence, model interpretability, hyperparameter tuning, computational resource requirements, and generalization ability cannot be ignored. SUMMARY

[0006] The technical problem to be solved by the present application is to provide a structure seismic response calculation method based on dynamic guidance support vector machine, which solves the obvious shortcomings of the existing classical structure seismic response calculation method in stability, accuracy and sensitivity to time step selection, and the two key problems of low data dependence and low model interpretability of traditional machine learning models.

[0007] To solve the above technical problems, the technical solution adopted by the present application is a structure seismic response calculation method based on dynamic guidance support vector machine, which comprises the following steps:

[0008] Step 1: Establish a finite element model for dynamic analysis of the structure, extract the dynamic characteristics of the structure as the input of the model;

[0009] Step 2: According to the dynamic characteristics of the structure, establish a target function, solve the target function to select the optimal parameters of the model, and realize the optimization of the prediction model parameters;

[0010] Step 3: Based on the LS-SVM algorithm, taking the properties of the structure system and the ground motion as input and the displacement, velocity and other responses as output, a PDLS-SVM model is established;

[0011] Step 4: Use modal decomposition to decouple the multi-degree-of-freedom system. Based on the dynamic response of each mode, apply the principle of modal superposition to establish a seismic response prediction model for the structure.

[0012] Step 5: Based on the linear interpolation recursive formula, examine the performance of the seismic response prediction model of the structure and obtain its performance evaluation index.

[0013] In the preferred embodiment, Step 1 extracts the dynamic characteristics of the structure, including the core parameters of structural dynamics analysis such as the mass matrix, stiffness matrix, natural frequency, period, and modal shape of the structure; the finite element model is derived from the PDLS-SVM model analysis of the designed cantilever column with lateral degrees of freedom as a single-degree-of-freedom system and the seven-story three-dimensional frame structure model as a multi-degree-of-freedom system.

[0014] In the preferred embodiment, the theoretical basis of the LS-SVM algorithm in Step 3 is based on statistical learning theory and optimization theory. It solves linear system problems by using the sum of squared errors as the loss function and equality constraints instead of inequality constraints.

[0015] In the preferred embodiment, the theoretical basis of the PDLS-SVM model in Step 3 is the combination of the structural dynamics characteristics of the finite element model and the classical LS-SVM method. It is implemented through a dynamic principle-driven approach, using the equations of motion of a single-degree-of-freedom system as constraints to determine the parameters of the prediction model. Its mathematical principle can be expressed as follows:

[0016] Given the equations of motion of a linear single-degree-of-freedom (SDOF) system under ground motion, where :

[0017] (1)

[0018] Its initial conditions are:

[0019]

[0020] In the formula, m represents the mass matrix of the single-degree-of-freedom system, c is the damping matrix, k is the stiffness matrix, and u is the displacement vector. It is a velocity vector. It is an acceleration vector.

[0021] Equation (1) controls the motion of a linear single-degree-of-freedom system under the action of ground acceleration. Divide both sides of the equation by We can obtain:

[0022] (2)

[0023] In the formula, is the damping ratio of the nth mode shape; is the natural frequency of the damped vibration.

[0024] If the training samples are , , The basic idea of the classical LSSVM method is to use a nonlinear mapping to map the input data of the samples to a higher dimensional feature space, and then to perform optimal separation or fitting of the data in this high dimensional feature space, i.e. to determine the parameters of the prediction model by minimizing a regularized loss function The original form of the prediction model is , is the predicted response of the predictor Based on the principle of structural risk minimization, this problem can be described as solving the following optimization problem:

[0025] (3)

[0026]

[0027] wherein denotes the prediction error of the nth training sample; is a regularization parameter; is a feature vector. With reference to the classical LSSVM method, the seismic response predicted by the prediction model, is the predicted displacement response in time t; assuming that the high dimensional feature mapping function is sufficiently differentiable, given the ground motion in order to obtain the parameters of the prediction model, it can be equivalent to solve the following optimization problem:

[0028] (4)

[0029]

[0030] This optimization problem is solved using the Lagrange method:

[0031] (5)

[0032] wherein is a Lagrange multiplier, and according to the Karush_Kuhn_Tucker (KKT) condition, it can be obtained that:

[0033] (6)

[0034] (7)

[0035] (8)

[0036] (9)

[0037] (10)

[0038] (11)

[0039] The present application considers Gaussian kernel function RBF, which is ingenious in that it can implicitly calculate inner product in high-dimensional feature space without explicitly mapping data, so that originally linearly inseparable data becomes linearly separable in high-dimensional space, and its mathematical model is as follows:

[0040] (12)

[0041] (13)

[0042] In the formula, is a kernel parameter, and the inner product of the feature vector and is replaced by the kernel function , so that the matrix equation is obtained:

[0043] (14)

[0044] In formula (19), and are respectively:

[0045] (15)

[0046] (16)

[0047] (17)

[0048] (18)

[0049] (19)

[0050] By solving formula (15)~(19) and parameter , the matrix equation (14) is established, and the coefficient is obtained. Because the kernel trick is used, the original prediction model is replaced by the dual form of formula (20), and the coefficient can also be substituted into formula (20). Therefore, by using the obtained prediction model, the dynamic response can be calculated in the dual form, which is:

[0051] (20)

[0052] The velocity response and acceleration response can be calculated by taking the first and second order derivatives of equation (20), respectively:

[0053] (21)

[0054] (22).

[0055] In a preferred embodiment, the theoretical basis of the anti-seismic response prediction model in Step 4 is to decouple the multi-degree-of-freedom system using modal decomposition to obtain N independent single-degree-of-freedom equations corresponding to the natural vibration modes; according to the dynamic response of each vibration mode, the modal superposition principle is used to integrate the dynamic response of the multi-degree-of-freedom system. Its mathematical principle can be expressed as:

[0056] The motion equation of a linear multi-degree-of-freedom system with N degrees of freedom under the action of ground motion is given by: where :

[0057] (23)

[0058] where , and , are the mass matrix, damping matrix and stiffness matrix of the multi-degree-of-freedom system, respectively; u, ,

[0059] are the displacement vector, velocity vector and acceleration vector of the system, respectively.

[0060] (24)

[0061] where is the modal matrix, ; is the rth modal; is the rth modal coordinate; substituting equation (24) and its first and second order inverses into equation (23) to obtain the control equation:

[0062] (25)

[0063] where ; is the damping ratio of the rth mode; is the natural frequency of the damped vibration.

[0064] In a preferred scheme, the verification and performance evaluation in Step 5 are compared according to the results obtained by the linear interpolation recursive formula, and then compared and analyzed in terms of calculation efficiency and calculation accuracy with traditional integral methods such as the central difference method and the Newmark method; finally, the performance indicators of the prediction model, the determination coefficient R 2 , the root mean square error RMSE, and the mean absolute error MAE, are evaluated to verify the performance of the model.

[0065] The structural seismic response calculation method based on the dynamics-guided support vector machine provided by the application has the following beneficial effects:

[0066] 1. The structural seismic response calculation method based on the dynamics-guided support vector machine solves the problem that the structural seismic response method based on the traditional integral method is sensitive to the selection of time steps. Improper selection of time steps may cause stability problems or loss of precision. The PDLS-SVM model reduces the sensitivity to the selection of time steps through its algorithm design, thereby reducing the uncertainty and complexity in the calculation process.

[0067] 2. The structural seismic response calculation method based on the dynamics-guided support vector machine solves the problems of low calculation accuracy and long calculation time of the structural seismic response method based on the traditional integral method through the development of the PDLS-SVM model.

[0068] 3. The PDLS-SVM model is compared with common machine learning methods. The problem that machine learning methods rely on a large amount of data to train the model is solved through the physical driving method. The PDLS-SVM model directly introduces the constraints of dynamic balance and initial conditions in the model, effectively couples the physical law and the machine learning model, reduces the demand for actual data, and improves the generalization ability of the model. BRIEF DESCRIPTION OF DRAWINGS

[0069] The application will be further described below with reference to the accompanying drawings:

[0070] Figure 1 is the technical roadmap of the application. DETAILED DESCRIPTION

[0071] As Figure 1 shown, the structural seismic response calculation method based on the dynamics-guided support vector machine comprises the following steps:

[0072] Step 1: Establish a finite element model for structural dynamic analysis, and extract the dynamic characteristics of the structure as the input of the model.

[0073] Step2: According to the dynamic characteristics of the structure, a target function is established, and the optimal parameters of the model are selected by solving the target function to realize the optimization of the prediction model parameters.

[0074] Step3: Based on the LS-SVM algorithm, the properties of the structure system and the ground motion are taken as the input, and the displacement, velocity and other responses are taken as the output to establish the PDLS-SVM model.

[0075] Step4: The modal decomposition method is used to realize the decoupling of the multi-degree-of-freedom system, and according to the dynamic response of each mode shape, the modal superposition principle is used to establish the seismic response prediction model of the structure.

[0076] Step5: Based on the linear interpolation recursive formula, the performance of the seismic response prediction model of the structure is tested, and the performance evaluation index is obtained.

[0077] In this embodiment, the dynamic characteristics of the structure extracted in Step1 include the core parameters of structural dynamics analysis such as mass matrix, stiffness matrix, natural frequency, period and modal shape; the finite element model is derived from the PDLS-SVM model analysis of the designed cantilever column with transverse freedom as a single degree-of-freedom system and a seven-story three-dimensional frame structure model as a multi-degree-of-freedom system.

[0078] Further, the theoretical basis of the LS-SVM algorithm in Step3 is based on statistical learning theory and optimization theory, which solves the linear system problem by using the sum of squared errors as the loss function and replacing the inequality constraint with the equality constraint.

[0079] Further, the theoretical basis of the PDLS-SVM model in Step3 is the combination of the structural dynamics characteristics of the finite element model and the classical LS-SVM method, which is realized by the driving mode of the dynamic principle, and the motion equation of the single degree-of-freedom system is taken as the constraint condition to determine the prediction model parameters, and its mathematical principle can be represented as:

[0080] The motion equation of a given linear single degree-of-freedom system (SDOF) under ground motion is: :

[0081] (1)

[0082] Its initial conditions are:

[0083]

[0084] In the formula, m represents the mass matrix of the single degree-of-freedom system, c is the damping matrix, k is the stiffness matrix, u is the displacement vector, is the velocity vector, is the acceleration vector.

[0085] Equation (1) governs the motion of a linear single-degree-of-freedom system subjected to ground acceleration , dividing both sides of the equation by , we have

[0086] (2)

[0087] where is the damping ratio of the nth mode of vibration; is the natural frequency of damped vibration.

[0088] If the training samples are , , The basic idea of the classical LSSVM method is to use a nonlinear mapping to map the input data of the samples to a higher dimensional feature space, in which the data is optimally separated or fitted, i.e., the parameters of the prediction model are determined by minimizing a regularized loss function ; the original form of the prediction model is , is the predicted response of the predictor , based on the principle of structural risk minimization, this problem can be described as solving the following optimization problem:

[0089] (3)

[0090]

[0091] where denotes the prediction error of the nth training sample; is the regularization parameter; is the feature vector. Referring to the classical LSSVM method, the seismic response predicted by the model, is the predicted displacement response in time t; assuming that the high-dimensional feature mapping function is sufficiently differentiable, given the ground motion , in order to obtain the parameters of the prediction model, it is equivalent to solving the following optimization problem:

[0092] (4)

[0093]

[0094] Solve this optimization problem using the Lagrange method:

[0095] (5)

[0096] where, is the Lagrange multiplier, according to Karush_Kuhn_Tucker (KKT) conditions, we can get:

[0097] (6)

[0098] (7)

[0099] (8)

[0100] (9)

[0101] (10)

[0102] (11)

[0103] The present application considers the Gaussian kernel function RBF, which is ingenious in that it can implicitly calculate the inner product in the high-dimensional feature space without explicitly mapping the data, so that the originally linearly inseparable data becomes linearly separable in the high-dimensional space, and its mathematical model is as follows:

[0104] (12)

[0105] (13)

[0106] where, is the kernel parameter, and the inner product of the feature vectors and is replaced by the kernel function , so that the matrix equation is obtained:

[0107] (14)

[0108] In formula (19), and are respectively:

[0109] (15)

[0110] (16)

[0111] (17)

[0112] (18)

[0113] (19)

[0114] By solving formula (15)~(19) and the parameters The matrix equation (14) is established, and the coefficients Because of the use of the kernel trick, the original prediction model is replaced by the model of the dual form of equation (20), and the coefficients can also be substituted into equation (20). Therefore, using the obtained prediction model, the dynamic response can be calculated in the dual form, which is:

[0115] (20)

[0116] The velocity response and acceleration response can be calculated by taking the first and second order derivatives of equation (20), respectively:

[0117] (21)

[0118] (22).

[0119] Further, the theoretical basis of the anti-seismic response prediction model described in Step 4 is to use the modal decomposition method to decouple the multi-degree-of-freedom system, obtaining N independent single-degree-of-freedom equations corresponding to the natural modes; according to the dynamic response of each mode, the modal superposition principle is used to integrate the dynamic response of the multi-degree-of-freedom system. Its mathematical principle can be expressed as:

[0120] The motion equation of a linear multi-degree-of-freedom system with N degrees of freedom under the action of ground motion is given, where where :

[0121] (23)

[0122] In the formula, , respectively, represent the mass matrix, damping matrix and stiffness matrix of the multi-degree-of-freedom system; u, , are respectively the displacement vector, velocity vector and acceleration vector of the system.

[0123] The seismic response of the multi-degree-of-freedom system can be expressed as:

[0124] (24)

[0125] In the formula, is the modal matrix, ; is the rth modal; is the rth modal coordinate; Substitute equation (24) and its first and second order inverses into equation (23) to obtain the governing equation:

[0126] (25)

[0127] where, ; is the damping ratio of the rth mode of vibration; is the natural frequency of the damped vibration.

[0128] Further, the verification and performance evaluation in Step 5 are based on the results obtained from the linear interpolation recursive formula, and then compared with traditional integral methods such as central difference method and Newmark method in terms of calculation efficiency and accuracy. Finally, the performance indicators of the prediction model are evaluated: the coefficient of determination R 2 , root mean square error RMSE, and mean absolute error MAE, to verify the performance of the model. The specific results are shown in Table 1:

[0129] Table 1 Comparison of the performance of PDLS-SVM with the classical models

[0130] Table.1 Comparison of the performance of PDLS-SVM with the classical models

[0131]

[0132] Table 1 shows the comparison of the performance of PDLS-SVM model with the classical seismic response prediction methods (including average acceleration method, central difference method and linear acceleration method) in terms of R 2 (coefficient of determination) and RMSE(root mean square error) two key indicators. First, the R 2 value of PDLS-SVM model is as high as 0.9987, close to perfect prediction (R 2 =1 represents perfect prediction). In contrast, the R 2 values of other classical methods are much lower than that of PDLS-SVM, among which the R 2 value of average acceleration method is the lowest, only 0.3404, and the R 2The values are 0.7652 and 0.8195 respectively. This shows that the PDLS-SVM model can capture the relationship between the seismic response and the predicted variables with very high accuracy, and its prediction result is very close to the actual value, while the traditional method has a large deviation in describing this relationship. Secondly, from the perspective of RMSE, the performance of the PDLS-SVM model is also excellent. Its RMSE value is only 0.4315, which is much lower than the other three classical methods. The RMSE of the average acceleration method is the highest, reaching 9.7068, and the RMSE values of the central difference method and the linear acceleration method are 5.7913 and 5.0779 respectively. The RMSE measures the deviation between the predicted value and the actual value of the model, and the low RMSE value of the PDLS-SVM model means that its prediction error is small and the accuracy is high.

[0133] In summary, the structural seismic response calculation method based on the kinetic guide support vector machine provided by the present application is used to solve the obvious deficiencies of the existing classical structural seismic response calculation method in stability, accuracy and sensitivity to time step selection, as well as the two key problems that the traditional machine learning model has low data dependency and low model interpretability. The calculation efficiency and prediction performance of the established PDLS-SVM prediction model are verified by comparative analysis with classical seismic response prediction methods, which has important scientific guiding significance for improving the safety and reliability of building structures under the action of earthquakes.

Claims

1. A structural seismic response calculation method based on dynamic-guided support vector machine, characterized in that, Includes the following steps: Step 1: Establish a finite element model for structural dynamic analysis and extract the dynamic characteristics of the structure as input to the model; Step 2: Based on the dynamic characteristics of the structure, establish the objective function, solve the objective function to select the optimal parameters of the model, and optimize the parameters of the prediction model; Step 3: Based on the LS-SVM algorithm, using the structural system properties and ground motion as inputs and displacement and velocity response quantities as outputs, establish a dynamic-guided support vector machine-based PDLS-SVM model for predicting structural seismic response. Step 4: Use modal decomposition to decouple the multi-degree-of-freedom system. Based on the dynamic response of each mode, apply the principle of modal superposition to establish a seismic response prediction model for the structure. Step 5: Based on the linear interpolation recursive formula, examine the performance of the seismic response prediction model of the structure and obtain its performance evaluation index.

2. The structural seismic response calculation method based on dynamic-guided support vector machine according to claim 1, characterized in that: Step 1 extracts the dynamic characteristics of the structure, including the core parameters of structural dynamics analysis such as the mass matrix, stiffness matrix, natural frequency, period, and modal shape. The finite element model is derived from the design of a cantilever column with lateral degrees of freedom as a single-degree-of-freedom system and a seven-story three-dimensional frame structure model as a multi-degree-of-freedom system PDLS-SVM model for analysis.

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