A Bayesian system identification and model selection method for building structures based on adaptive sequential Monte Carlo

Abstract

The present invention belongs to the technical field of disaster prevention and mitigation of building structures, and discloses a Bayesian system identification and model selection method for building structures based on adaptive sequential Monte Carlo. The method of the present invention solves the Bayesian system identification and model class selection problems of building structures in a unified manner. The proposed new formula for the model class evidence value can efficiently calculate high-dimensional integrals that cannot be calculated by traditional methods. These formulas can respectively quantify the data fit and information gain of the model class to select the optimal model class, while ensuring the accuracy of the model without being too complicated. The present invention can efficiently solve the problems of Bayesian system identification and model class selection of real complex building structures, so as to accurately predict the structural behavior under different loads, and provide accurate models and theoretical support for disaster prevention and mitigation of building structures in the later stage.

Description

Technical Field

[0001] The present invention belongs to the technical field of disaster prevention and mitigation of building structures, and relates to an adaptive sequential Monte Carlo method, specifically a Bayesian system identification and model selection method based on adaptive sequential Monte Carlo, which is used to solve the Bayesian system identification and model class selection problems of building structures, so as to accurately predict the structural behavior under different loads, and provide accurate models and theoretical support for disaster prevention and mitigation of later building structures. Background Art

[0002] Accurate mathematical models are very important in all fields of science and engineering. In the field of civil engineering, it is necessary to understand the dynamic characteristics of the structure based on the building structure dynamic system model and predict the structural response under various excitations to ensure the safety of the structure. For complex building structures in actual engineering, it is necessary to search the high-dimensional parameter space to systematically select model categories and quantify uncertainties, so that the building structure model is not too simple to accurately reflect the true characteristics of the structure, and the structural model is not too complex to increase the amount of calculation and uncertainty.

[0003] Scholars have proposed many classic methods to solve the problem of structural dynamic system identification, but there are still certain limitations in applying these methods to real building structures (Haidarpour A, Tee KF. Finite Element Model Updating for Structural Health Monitoring. Structural Durability & Health Monitoring 2020; 14 (1): 1-17). Uncertainty quantification is a basic problem in the identification of building structure systems. The Bayesian method provides an effective way to model and quantify uncertainty probabilistically, but the traditional Bayesian method is computationally intensive, so a more efficient algorithm is needed (Beck JL and Katafygiotis LS. Updating models and their uncertainties. I: Bayesian statistical framework. Journal of Engineering Mechanics 1998; 124 (4): 455-61). In addition, the Bayesian system identification and model class selection of building structures require different methods to be carried out separately, and there is a lack of a unified theory to unify the two, which is very time-consuming in practical applications. At present, there is still a need for a unified Bayesian method that can solve the problems of complex building structure system identification and model classification selection.

[0004] Existing methods for identifying building structure systems are inefficient and cannot quantify uncertainty and the pros and cons of different models in detail. Specifically, there are the following problems: (1) In system identification based on the Bayesian probability framework, the mechanism for generating building structure parameter samples from complex probability density functions is unclear, and it is impossible to adaptively sample according to the characteristics of different regions in the high-dimensional parameter space, resulting in low parameter sampling efficiency; (2) There is a lack of model selection methods that can separately quantify the data fit and information gain of building structure models, making it impossible to comprehensively evaluate the pros and cons of different models; (3) Existing methods are mostly used for small structures under laboratory conditions, and cannot efficiently solve the Bayesian system identification and model class selection problems of real complex building structures and make accurate predictions of structural behavior.

[0005] The present invention provides an efficient and practical solution for the selection of the optimal model of building structures and the establishment of accurate models, and provides accurate models and theoretical support for the accurate prediction of structural behavior under different loads, as well as disaster prevention and mitigation of building structures in the later stage. This specification only takes the application background of Bayesian system identification and model class selection of a certain building structure as an example, and the present invention is also applicable to other building structures. Summary of the invention

[0006] The present invention proposes an adaptive sequential Monte Carlo method, which solves the problem that traditional methods cannot uniformly identify building structure systems and select model types, and the calculation efficiency is greatly improved; it solves the problem that traditional methods cannot calculate high-dimensional integrals in model evidence values, and provides a quantitative method for the degree of fitting of building structure models to test data and the degree of information extracted from test data, selects the optimal structural model type, and identifies the precise model of complex building structures.

[0007] The technical solution of the present invention:

[0008] A Bayesian system identification and model selection method based on adaptive sequential Monte Carlo integrates Bayesian system identification and Bayesian model class selection into the adaptive sequential Monte Carlo method to identify the stiffness parameters of the building structure and select its optimal model class. This technical solution is implemented by sampling the posterior probability density of the structural stiffness parameters. After the sampling is completed, the optimal model class and accurate structural model of the building structure can be obtained. This solution innovatively proposes a multi-level adaptive sampling method, taking the bridge probability density function p connecting each level i for:

[0009]

[0010] Where c i is the normalization constant; J(θ) quantifies the error between the experimental and predicted modal parameters, and its specific expression is given in equation (1); is the variance parameter of the bridge probability density function.

[0011] The steps of the multi-level adaptive sampling method are as follows:

[0012] Step 1: Initialization

[0013] Let the sampling level i = 1; set the variance parameter of the bridge probability density function By randomly sampling, we obtain a sample of the building structure stiffness parameter θ, and substitute the sample into equation (1) to obtain J(θ). Among them, the expression of J(θ) is as follows:

[0014]

[0015] In the formula and are the natural frequency and vibration shape of the mth mode obtained from the experiment; ω m (θ) and 𝛟 m (θ) are the natural frequency and vibration shape of the mth mode predicted by the model; N m The number of available experimental modes; set the number of samples generated per layer N s ; Use Metropolis-Hastings (MH) algorithm to calculate the building structure stiffness parameter samples Sampling is performed, and the sampling probability density function is set to uniform distribution, and the target probability density function is the first-layer bridge probability density function p1; the MH algorithm assigns equal weight to each sample, that is,

[0016] Step 2: Update the next level i=i+1

[0017] Bridging probability density function p i The variance parameter By minimizing equation (2), we can obtain

[0018]

[0019] Among them, ESS is the number of effective samples, which is used to quantify whether the generated samples are equally important; α is the ESS reduction factor, which is taken as 0.99; is the normalized weight, satisfying Calculated by equation (4):

[0020]

[0021] In the formula, In the sample The weight increment at ;

[0022] Step 3: Sample at the current level i, using the Markov chain Monte Carlo kernel based on the MH algorithm; for j = 1 to N s ,

[0023] (1) From the previous layer of samples Take a sample: in, Representative samples The Dirac measure of

[0024] (2) From Select candidate samples X from the normal distribution centered at: in, The representative mean is The covariance matrix is ​​C i-1 The normal distribution probability density function, C i-1 It is used The estimated sample covariance matrix;

[0025] (3)X has The probability of being accepted is X has a 1-r probability of being rejected and the previous sample is adopted, that is, where q(·) is the sampling probability density function, which is a mixed Gaussian distribution with each sample in the previous level as the mean and the sample covariance matrix as its covariance matrix.

[0026] Step 4: Check termination criteria

[0027] if Then proceed to step 5; otherwise return to step 2;

[0028] Step 5: Based on kernel density estimation, use the final level samples to estimate the posterior probability density function of the building structure stiffness parameters:

[0029]

[0030] Among them, the experimental data set Including experimental natural frequencies and vibration mode N m N is the number of available experimental modes; L is the number of samples in the final layer; is the final layer N L The empirical covariance matrix estimated by samples;

[0031] Step 6: Select the optimal model class

[0032] Based on the generated building structure stiffness parameter samples, the present invention proposes a new method for selecting the optimal building structure model class, that is, given the same measurement data, the evidence values ​​of multiple candidate model types are calculated, and the model class with the maximum evidence value (that is, the optimal building structure model class) is selected from these candidate model types. The present invention proposes a new method for calculating the evidence value of the building structure model class, which solves the problem that the traditional method cannot independently quantify the data fit and information gain of the model class. The data fit reflects the accuracy of the building structure model in reproducing the experimental data, and the information gain reflects the model complexity of the building structure model in extracting information. By independently quantifying these two physical quantities, it is ensured that the building model structure is sufficiently accurate while not being too complex to cause excessive calculation and uncertainty. The data fit, information gain and model class evidence value of each model class are calculated using equations (6) to (8):

[0033]

[0034]

[0035] Where M k Refers to the model class; N L is the number of samples in the final layer; ln(p(θ|M k )) is independent of the sample and is a constant.

[0036] The beneficial effects of the present invention are as follows: (1) The present invention proposes a novel adaptive sampling method to solve the Bayesian system identification and model class selection problems of building structures in a unified manner; (2) The new formula of model class evidence value proposed according to the adaptive sequential Model Carlo framework can efficiently calculate high-dimensional integrals that cannot be calculated by traditional methods. These formulas can respectively quantify the data fit and information gain of the model class to select the optimal model class; (3) The Bayesian system identification and model class selection problems of actual complex engineering structures can be efficiently solved to accurately predict the structural behavior under different loads, and provide accurate models and theoretical support for disaster prevention and mitigation of later building structures.

[0037] This method constructs the posterior probability density function of complex structural parameters based on experimental data, and identifies the probability density function by efficiently generating parameter samples in high-dimensional parameter space, thereby selecting the optimal structural model class and accurately identifying it. This method can be used in fields such as probabilistic failure analysis, building automation, and structural health monitoring, and its advantage is high computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a flow chart of the present invention.

[0039] Figure 2 Provide finite element modeling of full-scale structures.

[0040] Figure 3 Comparison of predicted mode shapes (dashed lines) and experimental mode shapes (solid lines). DETAILED DESCRIPTION

[0041] In order to make the above features and advantages of the present invention easier to understand, a full-scale coupled structure system is taken as an example to perform system identification and optimal model type selection. The implementation steps of the present invention are described in detail below with reference to the accompanying drawings:

[0042] Finite element modeling of full-scale coupled structures Figure 2 Six model classes of the structure are selected, and each model class assigns different stiffness parameters to different columns of the structure to identify the lateral stiffness distribution of the structure and accurately reflect the dynamic characteristics of the structure.

[0043] The model selection is calculated according to Figure 1 The steps are as follows:

[0044] Step 1: Initialize the adaptive sequential Monte Carlo. Let the level index i = 1, obtain the stiffness parameters of the coupled structure by random sampling, and substitute them into equation (1) to obtain the square error between the finite element predicted modal parameters and the experimental modal parameters of the structure and the variance parameter σ1 as the first-level bridging probability density function: 2 .

[0045] Step 2: Update the variance parameter of the bridge probability density function and the weight of the stiffness parameter of the coupled structure by minimizing equation (2), and perform stiffness parameter sampling at the next level.

[0046] Step 3: Sampling at level i. Using the Markov Chain Monte Carlo kernel based on the MH algorithm. For j = 1 to N s ,

[0047] (1) Draw a sample from the structural stiffness parameter samples of the previous level, and the probability of each sample being drawn is the weight value of each sample;

[0048] (2) Select a sample from Select candidate samples X from the normal distribution probability density function centered on : in, Representative is the center and the covariance matrix is ​​C i-1 The normal distribution probability density function, C i-1 It is used The estimated empirical covariance matrix;

[0049] (3) accepting the candidate stiffness parameter sample X with a certain probability (see the specific steps of the technical solution for details), that is, if the stiffness parameter sample makes the structural modal parameters predicted by the finite element model closer to the experimental modal parameters, then the sample has a greater probability of acceptance, otherwise it has a greater probability of rejection;

[0050] Step 4: Check the termination criteria. If the variance parameter changes of the two levels are small enough, that is, the bridge density function converges to a certain parameter space region, proceed to step 5; otherwise, return to step 2.

[0051] Step 5: Based on kernel density estimation, the posterior probability density function is estimated using the final level structural stiffness parameter sample:

[0052]

[0053] Among them, the experimental data set Including experimental natural frequency and vibration mode N m N is the number of available experimental modes; L is the number of samples in the final layer; is the final layer N L The empirical covariance matrix estimated by the samples.

[0054] Step 6: Select the optimal model class for the coupling structure, and use equations (6) to (8) to calculate the data fit, information gain and model class evidence value of each coupling structure model class respectively.

[0055] The complexity of these 6 model classes increases from M1 to M6. The following table summarizes the data fitting, information gain and evidence value of each model class. It can be seen from the table that the method proposed by the present invention shows that, given the same set of experimental modal parameter data, different model classes considering different distributions of structural stiffness have different evidence values. The most complex model class is not necessarily the optimal model class, and the more complex the model class is, the better it can fit the data. For example, M6 is more complex than M5, but the data fitting of M5 is greater than that of M6, that is, M5 can fit the data better. At the same time, since M6 is more complex than M5, M6 extracts more information from the data, that is, the information gain value of M6 is greater than that of M5. According to the present invention, a larger information gain will reduce the evidence value when calculating the evidence value, limiting the complexity of the model, so the present invention ensures a balance between model accuracy and complexity. M4 has the largest evidence value (largest posterior probability) among these 6 model classes and is the optimal model class. Figure 3 The comparison between the predicted modal vibration shape (dashed line) and the experimental modal vibration shape (solid line) of this model class shows that the predicted model is very accurate, indicating that the present invention can identify a model that accurately reflects the actual dynamic characteristics of the building structure.

[0056]

Claims

1. A Bayesian system identification and model selection method for building structures based on adaptive sequential Monte Carlo, characterized in that: The Bayesian system identification and Bayesian model class selection are integrated into the adaptive sequential Monte Carlo method to identify the stiffness parameters of the building structure and select the optimal model class of the building structure. This is achieved by sampling the posterior probability density of the stiffness parameters of the building structure. After the sampling is completed, the optimal model class of the building structure and the accurate structural model can be obtained. A multi-level adaptive sampling method is proposed, taking the bridge probability density function p connecting each level i for: In the formula, c i is the normalization constant; J(θ) is the error between the quantified experimental and predicted modal parameters, and its specific expression is given in equation (1); is the variance parameter of the bridge probability density function; The steps of the multi-level adaptive sampling method are as follows: Step 1: Initialization Let the sampling level i = 1; set the variance parameter of the bridge probability density function A sample of the building structure stiffness parameter θ is obtained by random sampling, and the sample is substituted into equation (1) to obtain J(θ). Let Among them, the expression of J(θ) is as follows: In the formula, and are the natural frequency and vibration shape of the mth mode obtained from the experiment; ω m (θ) and 𝛟 m (θ) are the natural frequency and vibration shape of the mth mode predicted by the model; N m The number of available experimental modes; set the number of samples generated per layer N s ; Use MH algorithm to calculate the building structure stiffness parameter samples Sampling is performed, and the sampling probability density function is set to uniform distribution, and the target probability density function is the first-layer bridge probability density function p1; the MH algorithm assigns equal weight to each sample, that is, Step 2: Update the next level i=i+1 Bridging probability density function p i The variance parameter By minimizing equation (2), we obtain Among them, ESS is the number of effective samples, which is used to quantify whether the generated samples are equally important; α is the ESS reduction factor, which is taken as 0.99; is the normalized weight, satisfying Through equation (4), we can calculate: In the formula, In the sample The weight increment at ; Step 3: Sample at the current level i, using the Markov chain Monte Carlo kernel based on the MH algorithm; for j = 1 to N s , (1) From the previous layer of samples Take a sample: in, Representative samples The Dirac measure of (2) From Select candidate samples X from the normal distribution centered at: in, The representative mean is The covariance matrix is ​​C i-1 The normal distribution probability density function, C i-1 It is used The estimated sample covariance matrix; (3)X has The probability of being accepted is X has a 1-r probability of being rejected and the previous sample is adopted, that is, Where q(·) is the sampling probability density function, which is in the form of the mean of each sample in the previous level, and the sample covariance matrix is ​​the mixed Gaussian distribution of its covariance matrix; Step 4: Check termination criteria if Then proceed to step 5; otherwise return to step 2; Step 5: Based on kernel density estimation, use the final level samples to estimate the posterior probability density function of the building structure stiffness parameters: Among them, the experimental data set Including experimental natural frequencies and vibration mode N L is the number of samples in the final layer; is the final layer N L The empirical covariance matrix estimated by samples; Step 6: Select the optimal model class Based on the generated building structure stiffness parameter samples, a new method for selecting the optimal building structure model class is proposed. That is, given the same measurement data, the evidence values ​​of multiple candidate building structure model types are calculated, and the model class with the largest evidence value is selected from these candidate building structure model types, that is, the optimal building structure model class; the data fit reflects the accuracy of the building structure model in reproducing the experimental data, and the information gain reflects the model complexity of the building structure model in extracting information. By independently quantifying these two physical quantities, it is ensured that the building structure model structure is sufficiently accurate while not being too complex to cause excessive calculation and uncertainty; the data fit, information gain and model class evidence value of each model class are calculated respectively using equations (6) to (8): Where M k Refers to the model class; ln(p(θ|M k )) is independent of the sample and is a constant.

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