Adaptive model update method for probabilistic analysis of complex structures performed by computers

Abstract

This provides an adaptive model update algorithm for the probabilistic analysis of complex structures. [Solution] To provide guidance for the soundness diagnosis of complex structures, the method includes determining the parameter distribution and coefficients required for the algorithm to be corrected; sampling according to the prior distribution of parameters and calculating the likelihood value; adaptively calculating other algorithm coefficients and starting the first iteration; calculating the covariance matrix of the proposed distribution and generating intermediate model parameters; obtaining candidate values ​​based on the proposed distribution sampling; determining whether the candidate values ​​are acceptable; updating the internal coefficients and completing one iteration after all candidate values ​​have been determined; determining whether the stopping condition is met, and if so, iterating again to generate the posterior distribution of the model parameters; otherwise, returning and running a new iterative round.

Description

[Technical Field]

[0001] The present invention relates to the technical field of model structure parameter inversion, and more particularly to an adaptive model update algorithm for the probabilistic analysis of complex structures. [Background technology]

[0002] Performing parameter inversion from test data of complex structures to diagnose their structural integrity is an economical and reliable method, providing significant guidance for the safety assessment of complex structures. Complex structures refer to large-scale and complex engineering structures such as high-rise and super-high-rise buildings, large-span bridges, and nuclear power plants. Due to the large size and numerous components of complex structures, testing them, even using scale models, is extremely time-consuming and costly. Furthermore, studying the safety of complex structures typically requires applying loads to the model structure until failure. Established models only provide one set of structural failure data, and repeated testing places a significant burden on human, material, and financial resources. As a result, completely inverting the parameter values ​​of complex structures through testing is extremely difficult, hindering the advancement of research on the structural integrity assessment of complex structures. Moreover, complex structures have numerous parameters that need to be verified. While some material parameters can be verified through material property testing, many cannot be verified through testing or specifications. Due to uncertainties in the parameters themselves, measurement errors, and environmental conditions, verifying the parameters of complex structures is more difficult, and it is necessary to consider parameter uncertainties when analyzing complex structures. Finally, when constructing a complex engineering structure model using the finite element method and parameter inverse analysis and performing parameter inverse analysis, the long computation time for model parameter inversion and the applicability of the method to complex engineering models are issues that need to be considered. Therefore, establishing a highly adaptable model update method that uses a small amount of data for complex structures is a crucial step in solving the parameter inversion and soundness diagnosis of complex structures.

[0003] Bayesian model parameter updating is one of the most popular model parameter updating methods, and it is a model parameter modification method based on Bayesian theory and the Markov Chain Monte Carlo (MCMC) sampling algorithm. Since the 21st century, Bayesian model parameter updating algorithms and finite element models have been widely used for model parameter inversion and health assessment, and the most popular Bayesian model parameter updating method is based on the Metropolis-Hastings (MH) sampling algorithm. Due to the constraints of the MH sampling algorithm, several new and improved algorithms have been proposed, such as the transition MCMC (TMCMC) algorithm, the delayed rejection MCMC (DR-MCMC) algorithm, and the adaptive Metropolis-MCMC (AM-MCMC) algorithm. However, these algorithms have certain limitations. First, no suitable likelihood function calculation method has been proposed for different models, so each time the algorithm is applied to a new model, it is necessary to find the appropriate form of the likelihood function again. Second, the algorithm cannot adaptively find the correct values ​​for some coefficients, so it takes a lot of time to adjust these coefficients. Finally, the problem of long computation times for finite elements is not considered, and the algorithm requires multiple iterations to solve some algorithm parameters. These methods may be suitable for models with short iteration times each time, but are not applicable when the iteration times are long. [Overview of the project]

[0004] The present invention aims to provide an adaptive model update algorithm for the probabilistic analysis of complex structures. [Means for solving the problem]

[0005] The present invention (1) i is the total number of iterations of the algorithm, and N is the number of iterations in each iteration. s Let c be the difference constant, α be the exponential constant, g be the likelihood function, and let θ = [θ1, θ2, ..., θ] be the matrix of important parameters of the model.ii ,…, θ n (where \(i = 1, 2, \ldots, n\) and \(\theta\) ii represents the \(i\)-th important parameter), the prior distribution of the model parameters is \(\pi(\theta)\), the number of measurement points is \(N\) y , the measurement point response matrix of the test is \(Y = [y_1, y_2, \ldots, y\) jj ,…, y Ny (where \(j = 1, 2, \ldots, N\) y and \(y\) jj represents the response of the \(j\)-th measurement point), and set \(q_0 = 0\), (2) Set \(i = 1\), and randomly sample \(N\) s sets of prior parameter matrices \(\Theta = [\theta\) (1) , \theta\) (2) ,…, \theta\) (j) ,…, \theta\) (Ns) (where \(j = 1, 2, \ldots, N\) s and \(\theta\) (j) represents the \(j\)-th set of prior parameter matrices), and calculate the likelihood value of the prior parameters of the corresponding model, (3) Calculate the variance constant \(C\) cov , \(q\) i and the maximum value \(C\) max of the difference, and the likelihood weight coefficient \(w\) (i,j) of each set of prior parameter values and the average \(S\) i of the weight coefficients at the \(i\)-th total iteration number, (4) Calculate the variance matrix \(\Sigma\) i of the proposed distribution, (5) Generate an intermediate model parameter matrix \(\Theta\) c = [\theta\) (1,c) , \theta\) (2,c) ,…, \theta\) (j,c) ,…, \theta\) (Ns,c) (where \(j = 1, 2, \ldots, N\) s and \(\theta\) (j,c) represents the \(j\)-th set of intermediate model parameter matrices), and randomly sample based on the intermediate parameter matrix and the variance matrix \(\Sigma\) i of the proposed distribution to obtain \(N\) s sets of candidate values \(\Theta\) cc = [\theta\)(1,cc) ,θ (2,cc) ,…,θ (j,cc) ,…,θ (Ns,cc) ](j=1, 2, ..., N s And so, θ (j,cc) (where represents the j-th candidate value matrix), and θ (j,cc) If it exceeds the range of the prior distribution, Σ i It is reduced by a factor of 2 and resampled to θ (j,cc) Finally, we obtain the likelihood value g(Y|θ) of the intermediate parameter and candidate value. (j,c) ) and g(Y|θ (j,cc) The steps to calculate each of the following: (6) Randomly generate a value u from [0,1], and u≦(g(Y|θ) (j,cc) )) / (g(Y|θ (j,c) )) in the case of θ (j) =θ (j,cc) If not, θ (j) =θ (j,c) The step, (7)C max , g, w (i,j) and S i Steps to update, and (8) Let i = i + 1, q i-1 If ≥ 1, q i-1 Set = 1, repeat steps (4) to (7) to start a new iteration round, calculate S, then the iteration ends, q i-1 If ≤ 1, the process includes returning to step (4).

[0006] Furthermore, in step (1) above, the number of iterations each time is N s The coefficient of the function is 100, the difference constant c is ln(60)-ln(70), the exponential constant α is 0.06-0.1, the important parameter matrix θ is determined by selecting parameters that significantly influence the model through sensitivity analysis, the prior distribution π(θ) of the model parameters is obtained from industry standards, journal articles, and test data, the number of sensors and the response of the measurement points depend on the test data and the number of measurement points of interest to the user, and the form of the likelihood function is expressed as follows.

[0007]

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[0008] Furthermore, the variance constant in step (3) above is expressed as follows:

[0009]

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[0010] Furthermore, the maximum value of the difference in step (3) above is expressed as follows:

[0011]

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[0012] Furthermore, the likelihood weight coefficient w of the prior parameter values ​​for each set in step (3) above. (i,j) It can be expressed as follows:

[0013]

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[0014] Furthermore, the average S of the prior parameter likelihood weight coefficients in step (3) i It can be expressed as follows:

[0015]

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[0016] Furthermore, the variance matrix Σ of the proposed distribution in step (4) above i It can be expressed as follows:

[0017]

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[0018] Furthermore, in step (7), after the iteration of each set is completed, the maximum difference C maxLikelihood function g, Likelihood weight w (i,j) and the average S of the likelihood weights i Update.

[0019] Furthermore, in step (8) above, q i-1 If ≥ 1, q i-1 Set = 1, and after one iteration, the posterior distribution of the obtained model parameters is calculated using the following formula to obtain the likelihood weight values.

[0020]

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[0021] Compared to prior art, the present invention has the following significant advantages. The present invention sets the values ​​of the number of iterations, difference constants, and exponential constants for each iteration, combines the proposed likelihood function to adaptively find the most likely distribution of structural model parameters, and provides further guidance for the soundness diagnosis of complex structures. Previous studies required obtaining appropriate algorithm coefficients through multiple iterations and lacked the constraint of a readily adaptable likelihood function shape. The present invention overcomes this constraint, comprehensively utilizing the data obtained during the iteration process to adaptively find algorithm coefficients and likelihood function shapes suitable for different complex engineering models, thereby improving the adaptability of model modifications, reducing computation time, and improving algorithm efficiency. [Brief explanation of the drawing]

[0022] [Figure 1] This is a flowchart of the present invention. [Figure 2] This is a finite element model of a prestressed concrete containment vessel. [Figure 3] This graph shows the change in the minimum value of the overall error across 30 measurement points during the iterative process. [Figure 4] This graph shows the histograms of each parameter after model modification and the fitting results. [Figure 5] This graph shows the error distribution of 30 measurement points during the iterative process. [Modes for carrying out the invention]

[0023] The technical means of the present invention will be described in detail below with reference to the attached drawings.

[0024] As shown in Figure 1, the present invention includes the following steps.

[0025] (1) i is the total number of iterations of the algorithm, and N is the number of iterations in each iteration. s Let c be the difference constant, α be the exponential constant, g be the likelihood function, and let θ = [θ1, θ2, ..., θ] be the matrix of important parameters of the model. ii ,…,θ n ](Here, ii = 1, 2, ..., n, θ ii (where is the second important parameter), the prior distribution of the model parameters is π(θ), and the number of measurement points is N. y The measurement point response matrix for the test is Y=[y1,y2,…,y jj ,…,y Ny ](Here, jj = 1, 2, ..., N y And so, y jj (where represents the response of the jj-th measurement point), and set q0=0 in the following step. i does not require user confirmation, and in step (1) above, the number of iterations N s The coefficient of the function is 100, the difference constant c is ln(60)-ln(70), the exponential constant α is 0.06-0.1, the important parameter matrix θ is determined by selecting parameters that significantly influence the model through sensitivity analysis, the prior distribution π(θ) of the model parameters is obtained from industry standards, journal articles, and test data, the number of sensors and the response of the measurement points depend on the test data and the number of measurement points of interest to the user, and the form of the likelihood function is expressed as follows.

[0026]

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[0027] (2) Set i=1 and take the prior distribution π(θ) of the model parameters N s The prior parameter matrix of the pair Θ=[θ (1) , θ (2) , ..., θ (j) , ..., θ (Ns) ](Here, j=1, 2, ..., N s And so, θ (j) The steps are to randomly sample the j-th set of prior parameter matrices (where represents the j-th set) and calculate the likelihood value of the corresponding prior parameter of the model, (3) Dispersion constant C cov , q i and the maximum difference C max , and the likelihood weight coefficient w of the prior parameter values ​​for each set in the i-th total iteration. (i,j) and the average S of the weight coefficients i Steps to calculate The variance constant is expressed as follows:

[0028]

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[0029] The maximum difference can be expressed as follows:

[0030]

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[0031] Likelihood weight coefficient w of the prior parameter values ​​for each pair (i,j) It can be expressed as follows:

[0032]

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[0033] The mean S of the prior parameter likelihood weight coefficients i It can be expressed as follows:

[0034]

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[0035] (4) The covariance matrix Σ of the proposed distribution i Calculating step, The covariance matrix Σ of the proposed distribution i Is expressed as follows.

[0036]

Equation

[0037] (5) The likelihood weight coefficient of the prior parameter value and the intermediate model parameter matrix Θ c =[θ (1,c) , θ (2,c) , …, θ (j,c) , …, θ (Ns,c) (where j = 1, 2, …, N s and θ (j,c) represents the j-th intermediate model parameter matrix) is generated, and random sampling is performed based on the intermediate parameter matrix and the covariance matrix Σ of the proposed distribution i to obtain N s sets of candidate values Θ cc =[θ (1,cc) , θ (2,cc) , …, θ (j,cc) , …, θ (Ns,cc) (j = 1, 2, …, N s and θ (j,cc) represents the j-th candidate value matrix), and when θ (j,cc) exceeds the range of the prior distribution, Σ i is reduced by a factor of 2 and resampled to obtain θ (j,cc) , and finally the likelihood values g(Y|θ (j,c) ) and g(Y|θ (j,cc)This step involves calculating each of the following. When generating intermediate model parameters, if the proportion of the prior model parameters in the likelihood weights is large, the values ​​of the generated intermediate model parameters will be large. The number of intermediate model parameter values ​​that are the same as the prior model parameter values ​​is given by the following formula. Candidate values ​​are generated from a normal distribution with the intermediate model parameters as the mean and the variance matrix of the proposed distribution as the variance.

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[0038] (6) Randomly generate a value u from [0,1], and u≦(g(Y|θ) (j,cc) )) / (g(Y|θ (j,c) )) in the case of θ (j) =θ (j,cc) If not, θ (j) =θ (j,c) The step, (7)C max , g, w (i,j) and S i A step to update the maximum difference C after each set of iterations is complete. max Likelihood function g, Likelihood weight w (i,j) and the average S of the likelihood weights i Update.

[0039] (8) Let i = i + 1, q i-1 If ≥ 1, q i-1 Set = 1, and repeat steps (4) to (7) to start a new iteration round. After calculating TIFF2026047071000016.tif52, the iteration ends and q i-1 If ≤ 1, return to step (4). q i-1 If ≥ 1, q i-1 Set = 1, and after one iteration, the posterior distribution of the obtained model parameters is calculated using the following formula to obtain the likelihood weight values.

[0040]

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[0041] As an example of analysis, Figure 2 shows a finite element model of a prestressed concrete containment vessel. In this example analysis, there are five important parameters of the prestressed concrete containment vessel model, and their mean and variance distributions are shown in Table 1, which shows the parameter distribution of the prestressed concrete containment vessel.

[0042] [Table 1]

[0043] Also, the number of iterations in each iteration is N. s With =100, difference constant c=ln(70), and exponential constant α=0.10, the internal pressure of the containment vessel was increased from 0 MPa to 1.42 MPa, and 15 points were selected where the containment vessel reached loss of function (1.29 MPa) and structural failure (1.42 MPa) respectively, and the measurement point response was obtained. The file TIFF2026047071000019.tif52 was constructed, and the location distribution of the 15 measurement points and the numbers of the 30 measurement points are shown in Table 2.

[0044] [Table 2]

[0045] Once the above values ​​are defined, the algorithm automatically starts execution. The algorithm terminates after 7 iterations, and the overall error of the updated optimal parameters is 3.858 mm, which is 40.26% lower than the overall error of 6.458 mm before iteration. Figure 3 shows a graph showing the change in the minimum overall error of the 30 measurement points found in each iteration during the parameter iteration process. Figure 4 shows the histogram of each parameter and the fitting results after model modification. Taking the 95% confidence interval of each parameter distribution obtained in the fitting in Figure 4, 100 sets of parameter values ​​were randomly selected, and the error distribution diagram of the 30 measurement points was obtained as shown in Figure 5.

[0046] In short, this invention is based on a prior distribution of parameters, and after iterating the algorithm, the error of the found optimal parameters was approximately 40% lower than before the update. The mean of the error of randomly generated measurement points in the updated posterior distribution of parameters is also close to zero, which demonstrates the efficiency of the algorithm for updating parameters of complex structural models and can subsequently provide guidance for the health diagnosis of complex structures.

Claims

1. An adaptive model update algorithm for the probabilistic analysis of complex structures, (1) Let the total number of iterations of the algorithm be \(i\), the number of iterations each time be \(N\). s , the difference constant be \(c\), the exponential constant be \(\alpha\), the likelihood function be \(g\), the important parameter matrix of the model be \(\theta = [\theta １ , \theta ２ , \cdots, \theta ii , \cdots, \theta n \) (where \(i = 1, 2, \cdots, n\), and \(\theta ii \) represents the \(i\)-th important parameter), the prior distribution of the model parameters be \(\pi(\theta)\), the number of measurement points be \(N y , the response matrix of the measurement points in the test be \(Y = [y １ , y ２ , \cdots, y jj , \cdots, y Ny \) (where \(j = 1, 2, \cdots, N y , and \(y jj \) represents the response of the \(j\)-th measurement point), and let \(q ０ = 0\) step. (2) Set i = 1 and from the prior distribution π(θ) of the model parameters N s The prior parameter matrix of the pair Θ = [θ (１) θ (２) , ..., θ (j) , ..., θ (Ns) ] (where j = 1, 2, ..., N s And so, θ (j) The steps are to randomly sample the j-th set of prior parameter matrices (where represents the j-th set) and calculate the likelihood value of the corresponding prior parameter of the model, (3) Dispersion constant C ｃｏｖ , q i and the maximum difference C ｍａｘ , and the likelihood weight coefficient w of the prior parameter values ​​for each set in the i-th total iteration. (i，j) and the average S of the weight coefficients i Steps to calculate (4) Variance matrix Σ of the proposed distribution i Steps to calculate (5) Intermediate model parameter matrix Θ based on the likelihood weight coefficients of the prior parameter values ​​and the prior parameters. c = [θ (１，c) , θ (２，c) , ..., θ (j，c) , ..., θ (Ns，c) ] (where j = 1, 2, ..., N s And so, θ (j，c) The matrix is ​​generated (where represents the j-th set of intermediate model parameters), and the intermediate parameter matrix and the variance matrix Σ of the proposed distribution are generated. i Random sampling is performed based on N s Candidate values ​​Θ for the pair cc = [θ (１，cc) , θ (２，cc) , ..., θ (j，cc) , ..., θ (Ns，cc) ] (j = 1, 2, ..., N s And so, θ (j，cc) (where represents the j-th candidate value matrix), and θ (j，cc) If it exceeds the range of the prior distribution, Σ i It is reduced by a factor of two and resampled to θ (j，cc) Finally, the likelihood value g(Y|θ) of the intermediate parameter and candidate value is obtained, and finally the likelihood value of the intermediate parameter and candidate value is obtained. (j，c) ) and g(Y|θ (j，cc) The steps to calculate each of the following: (6) Randomly generate a value u from [0,1], and u ≤ (g(Y|θ) (j，cc) ) / (g(Y|θ) (j，c) )) in the case of θ (j ) = θ (j，cc) If not, θ (j) = θ (j，c) The step, (7) C ｍａｘ , g, w (i，j) and S i Steps to update, and (8) Let i = i + 1, q i-１ If ≥ 1, q i-１ Set = 1, repeat steps (4) to (7) to start a new iteration round, calculate S, then the iteration ends, q i-１ If ≤ 1, the step includes returning to step (4), In step (1) above, the number of each iteration N s An adaptive model update algorithm for the probabilistic analysis of complex structures, characterized in that the variance constant c is 100, the difference constant c is ln(60)-ln(70), the exponential constant α is 0.06 to 0.1, the important parameter matrix θ is determined by selecting parameters that greatly influence the model through sensitivity analysis, the prior distribution π(θ) of the model parameters is obtained from industry standards, journal articles, and test data, the number of sensors and the response of the measurement points depend on the test data and the number of measurement points of interest to the user, and the form of the likelihood function and the variance constant of step (3) are expressed as follows, respectively. [Math 1] [Math 2]

2. The adaptive model update algorithm for probabilistic analysis of complex structures according to claim 1, characterized in that the maximum value of the difference in step (3) is expressed as follows. [Math 3]

3. Likelihood weight coefficient w of each set of prior parameter values ​​in step (3) above (i，j) The adaptive model update algorithm for probabilistic analysis of complex structures according to claim 1, characterized in that it is expressed as follows. [Math 4]

4. The average S of the prior parameter likelihood weight coefficients in step (3) above i The adaptive model update algorithm for probabilistic analysis of complex structures according to claim 1, characterized in that it is expressed as follows. [Math 5]

5. The variance matrix Σ of the proposed distribution in step (4) above i The adaptive model update algorithm for probabilistic analysis of complex structures according to claim 1, characterized in that it is expressed as follows. [Math 6]

6. In step (7) above, after the iteration of each set is completed, the maximum difference C ｍａｘ Likelihood function g, Likelihood weight w (i，j) and the average S of the likelihood weights i An adaptive model update algorithm for probabilistic analysis of complex structures according to claim 1, characterized by updating the model.

7. In the above step (8), q i-１ If ≥ 1, q i-１ The adaptive model update algorithm for probabilistic analysis of complex structures according to claim 1, characterized in that the value is set to 1, the iteration is repeated once, and the posterior distribution of the obtained model parameters is calculated using the following formula to obtain likelihood weight values. [Number 7] [In the formula, if i = mm at the end, q i-１ The iteration terminates when the value becomes ≥ 1.

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