CN110096805B - Bridge structure parameter uncertainty quantification and transfer method based on improved self-service method - Google Patents
Bridge structure parameter uncertainty quantification and transfer method based on improved self-service method Download PDFInfo
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Abstract
The invention relates to a structural parameter uncertainty quantification and transmission method based on an improved self-service method under limited observation data. And performing information diffusion estimation on the probability density function of each Bootstrap subsample, and generating a large number of improved Bootstrap subsamples by adopting an acceptance-rejection method, thereby quantifying uncertainty based on large sample data. Secondly, based on a response surface model, a change interval of the structural response is rapidly calculated, the influence degree of parameter uncertainty on the structural response is judged according to a defined interval sensitivity index, and the uncertainty of the response is quantified. The method can quantify the uncertainty of the civil engineering structure parameters under the conditions of unknown probability distribution and limited measured data through mathematical theory analysis and experiments, and calculate the influence of the parameter uncertainty on the structure response, and can be used for engineering structure damage diagnosis, health monitoring, structure analysis and optimization design.
Description
Technical Field
The invention relates to the technical field of engineering structure analysis, in particular to a structural parameter uncertainty quantification and transfer method based on an improved self-service method under limited observation data.
Background
Various errors or uncertainty factors are inevitably generated in the civil engineering structure in the links of manufacturing, processing, assembling, modeling and the like, so that the structural parameters have uncertainty, and further the structural response also has uncertainty. The traditional structural analysis adopts a deterministic model and a deterministic method, the influence of uncertain factors is ignored, and the reliability of an analysis result is greatly reduced. At the moment, the uncertainty structure analysis method which is used as the development and extension of the traditional structure design method has more practical engineering background and important theoretical significance and application value.
The accuracy and the authenticity of an uncertainty analysis result depend on a description mode of an uncertainty source to a great extent, the existing method has relatively few researches on a reasonable description method of the uncertainty source, and particularly the reasonable description of uncertainty parameters under the condition of limited measured data. For example, when the probability distribution function assumed in the Monte Carlo method does not meet the actual probability distribution function, a large error is generated, thereby affecting the judgment of the actual service performance of the structure. Secondly, when the number of samples is small, the determined probability distribution itself has a certain uncertainty. Therefore, it is very meaningful to develop an uncertainty parameter quantification and transfer analysis method that does not depend on the probability distribution function.
Disclosure of Invention
In view of the above, the present invention provides a method for quantifying and transmitting uncertainty of structural parameters based on an improved self-service method under limited observation data, which can quantify uncertainty of structural parameters of civil engineering, calculate the influence of uncertainty of parameters on structural response, and can be used for damage diagnosis, health monitoring and structural analysis and optimization of engineering structures.
The invention is realized by adopting the following scheme: a structure parameter uncertainty quantification and transfer method based on an improved self-service method under limited observation data specifically comprises the following steps:
step S1: performing improved Bootstrap sampling and uncertainty parameter quantification based on an information diffusion theory;
step S2: and carrying out uncertainty transfer analysis based on the response surface model.
Further, step S1 specifically includes the following steps:
step S11: suppose that there are 4-limit observed data X ═ X 1 ,x 2 ,...,x n ) Is a subsample from an unknown population F, θ (F) being the unknown parameter (e.g., mean and standard deviation) of the population; according to X ═ X 1 ,x 2 ,...,x n ) An empirical distribution function of the original sample is constructed, as shown in equation (1):
in the formula, x (1) ≤x (2) ≤...≤x (n) Is x 1 ,x 2 ,...,x n Statistics after arrangement from small to large; n represents a sample capacity, and k represents a data array serial number (k ═ 1, 2., n-1);
step S12: from F n (x) Extracting Bootstrap subsample X * ＝(x 1 * ,x 2 * ,...,x n * ) (ii) a Probability density function f (x) obeyed by Bootstrap subsamples according to information diffusion theory * ) Comprises the following steps:
in the formula, mu (x) * ) Is a diffusion function defined as [ - ∞, + ∞ [ ]]The form of the above Borel measurable function has a large influence on the estimation result of the probability density function f (x) of the parent; the constant delta n is the window width, and n is the sub-sample capacity;
determining mu (x) using normal information diffusion estimation * ) Probability density function f (x) to which its estimated uncertainty parameter is subject * ) Comprises the following steps:
in the formula (I), the compound is shown in the specification,
the window width is defined as the value of gamma is related to n, and sigma represents the standard deviation of the self-help sample;
step S13: according to an estimated probability density function f (x) * ) Generating pseudo random numbers by adopting an acceptance-rejection method, and selecting random number sequences of uniformly distributed sequences;
step S14: repeating the steps S12 to S13B times to obtain the statistic R of B improved Bootstrap subsamples j ** Therefore, the distribution and the relevant characteristic value of the unknown parameter theta are statistically estimated, and the uncertainty parameter is quantized.
Further, step S13 is specifically: first, a distribution g (x) that is easy to sample is selected as the proposed distribution, and then a constant M > 1 is determined, so that f (x) Mg (x) is established in the definition domain of x; then generating a suggested random number y subject to a probability density function g (x) and a random number U subject to a uniform distribution U (0, 1); finally, a probability function of the reception criterion is calculated
If u < h (y), receiving the generated random number y; otherwise, discarding the random number y, and generating the random number again; the steps are cycled to obtain a plurality of improved Bootstrap subsamples X obeying the probability density function f (X) ** ＝(x 1 ** ,x 2 ** ,...,x m ** ) M > n, calculating the statistic R of the improved Bootstrap subsample ** (mean of improved Bootstrap subsamples
And standard deviation of
)。
Further, step S2 specifically includes the following steps:
step S21: modeling an ACE response surface;
step S22: the process of uncertainty parameter transfer analysis is illustrated with one parameter as an example. Firstly, B improved Bootstrap subsamples (each improved Bootstrap subsample has m data) of an uncertain parameter are respectively substituted into an ACE response surface, the average value of the other uncertain parameters is obtained, and the structural response corresponding to the improved Bootstrap subsamples is rapidly calculated; secondly, respectively calculating confidence intervals of the improved Bootstrap subsamples and the corresponding responses thereof under the condition that the confidence degree is 95 percent to obtain B improved Bootstrap subsamples and corresponding response intervals, taking the average value of the B confidence intervals as the value intervals of the uncertainty parameters and the corresponding structural responses thereof, and substituting the value intervals into the following formula to obtain the sensitivity factor of each parameter
In the formula (I), the compound is shown in the specification,
representing model parameters x i Induced structural response x i The amount of change of (c);
repeating the step S22, calculating the sensitivity factors of the other parameters, and judging the influence degree of the variation of each parameter on the uncertainty of the structural dynamic response by comparing the sensitivity factors of each parameter;
step S23: substituting the B group of improved Bootstrap subsamples with higher sensitivity factors and uncertain parameters into an ACE response surface, taking the average value of the parameters with lower sensitivity factors, calculating the B group of dynamic responses corresponding to the B group of improved Bootstrap subsamples, and further obtaining the statistical characteristic values of the B group of responses, such as the average value and the standard deviation; the distribution of the B statistics is used for simulating the distribution of the structural dynamic response mean value and the standard deviation, so that the sampling distribution and the distribution parameters of the response probability statistical characteristic value are obtained, and the effects of a plurality of parameter variations on the structural dynamic response variation and the estimation error of the structural response probability statistical characteristic generated by random sampling are estimated.
Further, step S21 is specifically: in order to improve the calculation efficiency of the uncertainty transfer analysis, a response surface model expressing the relationship between parameters and responses is fitted. Considering that the conventional parametric regression method usually specifies the form of the response surface and then performs regression, a large error may be generated when a response surface form that is not the best fit is specified. To this end, an alternating conditional expectation Algorithm (ACE) is introduced which does not require the prior assumption of an expression for the response surface model, but rather estimates by iteration the optimal nonlinear transformation φ of the independent x and dependent y variables 1 (x 1 ),φ 2 (x 2 ),...,φ n (x n ) And θ (y), and maximizes the statistical correlation therebetween. For structural response y and model parameters x 1 ,x 2 ,...,x i The multiple regression problem of (1) using an expression of an ACE regression model is as follows:
where θ (y) is a transformation function of the structural response y; phi is a i (x i ) A transformation function of the model parameters; ε represents the fitting error; after the transformation function is obtained, the expression of the response y is expressed by the following formula, namely the optimal response surface model:
in the formula, Θ (·) represents an inverse function of θ.
Compared with the prior art, the invention has the following beneficial effects: the method solves the problem that when the capacity of an original sample is small, Bootstrap subsamples obtained by resampling by a Bootstrap method are possibly very similar to the original sample, so that the calculation result deviates from the real distribution, can quantify the uncertainty of the civil engineering structure parameters and calculate the influence of the uncertainty of the parameters on the structure response under the conditions of unknown overall probability distribution and limited observation data, and can be used for engineering structure damage diagnosis, health monitoring, structure analysis and optimal design.
Drawings
FIG. 1 is a schematic flow chart of a method according to an embodiment of the present invention.
FIG. 2 is a diagram of a finite element model according to an embodiment of the present invention.
Fig. 3 is a mapping relationship diagram of the 1 st order natural frequency and uncertainty parameters and their corresponding ACE transformations according to an embodiment of the present invention.
FIG. 4 shows the results of the first 5 th order natural frequency sensitivity analysis (proposed method) according to the embodiment of the present invention.
FIG. 5 shows the result of the first 5 th order natural frequency sensitivity analysis (ANOVA) according to an embodiment of the present invention.
Detailed Description
The invention is further explained by the following embodiments in conjunction with the drawings.
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
As shown in fig. 1, the present embodiment provides: a structure parameter uncertainty quantification and transmission method based on an improved self-service method under limited test data specifically comprises the following steps:
step S1: carrying out improved Bootstrap sampling and uncertainty parameter quantification based on an information diffusion theory;
step S2: uncertainty transfer analysis based on the response surface model.
In this embodiment, step S1 specifically includes the following steps:
step S11: suppose that there are 4-limit observed data X ═ X 1 ,x 2 ,...,x n ) Is a subsample from an unknown population F, θ (F) being the unknown parameter (e.g., mean and standard deviation) of the population; according to X ═ X 1 ,x 2 ,...,x n ) An empirical distribution function of the original sample is constructed, as shown in equation (1):
in the formula, x (1) ≤x (2) ≤...≤x (n) Is x 1 ,x 2 ,...,x n Statistics after arrangement from small to large; n represents a sample capacity, and k represents a data array serial number (k ═ 1, 2.., n-1);
step S12: from F n (x) Extracting Bootstrap subsample X * ＝(x 1 * ,x 2 * ,...,x n * ) (ii) a Probability density function f (x) obeyed by Bootstrap subsamples according to information diffusion theory * ) Comprises the following steps:
in the formula, mu (x) * ) Is a diffusion function defined as [ - ∞, + ∞ [ ]]The above Borel measurable function has a form which has a large influence on the estimation result of the probability density function f (x) of the parent; the constant delta n is the window width, and n is the sub-sample capacity;
determining mu (x) using normal information diffusion estimation * ) Probability density function f (x) to which its estimated uncertainty parameter is subject * ) Comprises the following steps:
in the formula (I), the compound is shown in the specification,
the window width is defined as the value of gamma is related to n, and sigma represents the standard deviation of the self-help sample;
step S13: according to the estimated probability density function f (x) * ) Generating pseudo random numbers by adopting an acceptance-rejection method, and selecting random number sequences of uniformly distributed sequences;
step S14: repeating the steps S12 to S13B times to obtain the statistic R of B improved Bootstrap subsamples j ** Therefore, the distribution and the relevant characteristic value of the unknown parameter theta are statistically estimated, and the uncertainty parameter is quantized.
In this embodiment, step S13 specifically includes: first, a distribution g (x) that is easy to sample is selected as the proposed distribution, and then a constant M > 1 is determined, so that f (x) Mg (x) is established in the definition domain of x; then generating a suggested random number y subject to a probability density function g (x) and a random number U subject to a uniform distribution U (0, 1); finally, a probability function of the reception criterion is calculated
If u < h (y), receiving the generated random number y; otherwise, discarding the random number y, and generating the random number again; the steps are cycled to obtain a plurality of improved Bootstrap subsamples X obeying the probability density function f (X) ** ＝(x 1 ** ,x 2 ** ,...,x m ** ) M > n, calculating the statistic R of the improved Bootstrap subsample ** (mean of improved Bootstrap subsamples
And standard deviation of
)。
In this embodiment, step S2 specifically includes the following steps:
step S21: modeling an ACE response surface;
step S22: the process of uncertainty parameter transfer analysis is illustrated by taking one parameter as an example. Firstly, B improved Bootstrap subsamples (each improved Bootstrap subsample has m data) of an uncertain parameter are respectively substituted into an ACE response surface, the average value of the other uncertain parameters is obtained, and the structural response corresponding to the improved Bootstrap subsamples is rapidly calculated; secondly, respectively calculating confidence intervals of the improved Bootstrap subsamples and the corresponding responses thereof under the condition that the confidence degree is 95 percent to obtain B improved Bootstrap subsamples and corresponding response intervals, taking the average value of the B confidence intervals as the value intervals of the uncertainty parameters and the corresponding structural responses thereof, and substituting the value intervals into the following formula to obtain the sensitivity factor of each parameter
In the formula (I), the compound is shown in the specification,
representing model parameters x i The amount of change in the induced structural response y;
repeating the step S22, calculating the sensitivity factors of the other parameters, and judging the influence degree of the variation of each parameter on the uncertainty of the structural dynamic response by comparing the sensitivity factors of each parameter;
step S23: substituting the B group improved Bootstrap subsamples with higher sensitivity factors and uncertain parameters into an ACE response surface, taking the average value of the parameters with lower sensitivity factors, calculating the B group dynamic response corresponding to the B group improved Bootstrap subsamples, and further obtaining the statistical characteristic values of the B group response, such as the average value and the standard deviation; the distribution of the B statistics is used for simulating the distribution of the structural dynamic response mean value and the standard deviation, so that the sampling distribution and the distribution parameters of the response probability statistical characteristic value are obtained, and the effects of a plurality of parameter variations on the structural dynamic response variation and the estimation error of the structural response probability statistical characteristic generated by random sampling are estimated.
In this embodiment, step S21 specifically includes: in order to improve the calculation efficiency of the uncertainty transfer analysis, a response surface model expressing the relationship between parameters and responses is fitted. Considering that the conventional parametric regression method usually specifies the form of the response surface and then performs regression, a large error may be generated when a response surface form that is not the best fit is specified. To this end, an alternating conditional expectation Algorithm (ACE) is introduced which does not require the prior assumption of an expression for the response surface model, but rather estimates by iteration the optimal nonlinear transformation φ of the independent x and dependent y variables 1 (x 1 ),φ 2 (x 2 ),...,φ n (x n ) And θ (y), and maximizes the statistical correlation therebetween. For structural response y and model parameters x 1 ,x 2 ,...,x i The multiple regression problem of (1) adopts an expression of an ACE regression model as follows:
where θ (y) is a transformation function of the structural response y; phi is a i (x i ) A transformation function of the model parameters; ε represents the fitting error; after the transformation function is obtained, the expression of the response y is expressed by the following formula, namely the optimal response surface model:
in the formula, Θ (·) represents an inverse function of θ.
In particular, to examine the feasibility and reliability of the present embodiment, the distribution parameters of the population are estimated using a set of observed samples from a known population. It is known that original sample X ═ 1.1124,0.3679, -1.6876, -0.8223, -0.9069, -1.5200,1.6908, -3.0461, -0.3754, -0.3704, -0.1325,0.6321, -0.1546,0.3655,0.3436,0.7355,1.1339, -0.3106,1.2065, -0.3256] are independent random observed samples from a standard normal population N (0,1), and the number of samples N is 20.
And respectively adopting 3 methods of a traditional Bootstrap method, normal information diffusion and an improved Bootstrap method to estimate the standard deviation of the mean value of the group of samples. Table 1 gives the mean and standard deviation estimated by several methods. From table 1, it can be seen that: for small sample data, the error between the estimation result and the actual result of the traditional statistical analysis method is larger, the estimation result of the traditional Bootstrap method is improved compared with the traditional statistical method, and the average value and the standard deviation estimated by the improved Bootstrap method provided by the embodiment are closest to the actual value. Thus, it can be shown that the improved Bootstrap method has higher estimation accuracy of the overall distribution parameters in the case of small sample data than the conventional Bootstrap method.
Table 1 comparison of results for samples
In particular, this embodiment will be described in detail below.
1) Overview of the engineering
A certain bridge is a single-tower harp type certain prestressed concrete cable-stayed bridge with double cable surfaces, the span is 105m +97m, and a tower beam pier consolidation system is adopted. The main tower is a portal tower column, and the pier column part is a rectangular section. The basic section form of the main beam is a double-sided box, and the whole width of the main beam is 24 m. The full-bridge substructure adopts a bored pile foundation and a column pier.
According to a design drawing, an ANSYS is adopted to establish a space finite element model of the cable-stayed bridge, wherein a main Beam is simulated by a single ridge Beam with a self-defined box-shaped section, a bridge tower is simulated by a space Beam unit and Beam188, and a stay cable is simulated by a Link180 space rod unit. The full bridge has 2298 units and 4453 nodes. The initial finite element model is shown in FIG. 2.
The measured and calculated 4 th order natural frequencies of the main beam of the bridge are shown in table 2. As can be seen from table 2, the first 5 th order natural frequencies calculated based on the deterministic finite element model are all smaller than the measured values, and it can also be considered that the model parameters in the finite element model may not be consistent with the actual values of the bridge. This often happens in practical engineering, for example, the elastic modulus of concrete of in-service bridges is often about 10% greater than the nominal value.
TABLE 2 measured and calculated intrinsic frequency values
2) Frequency response surface model building
Modulus of elasticity E of main beam 1 Mass density M of main beam 1 Main tower elastic modulus E 2 Main column mass density M 2 Modulus of elasticity E of stay 3 Mass density M of stay cable 3 And cross-sectional area A of the inhaul cable with 3 cross-sectional forms 1 、A 2 、A 3 For inputting parameters, determining test points by adopting a uniform design method, and calculating the first 5-order natural frequency f corresponding to each test point 1 ～f 5 . And carrying out nonparametric regression by adopting an ACE algorithm, and establishing an ACE response surface of the first 5-order natural frequency. For example, the mapping relationship between the natural frequency of order 1 and each parameter and its corresponding ACE transformation is shown in fig. 3. The fitted correlation coefficients for the first 5 th order natural frequency ACE response surface are given in table 3. As can be seen from Table 3, the values of the fitting correlation coefficients for the first 5 natural frequencies are all close to 1, indicating that the ACE response surface of the fitted first 5 natural frequencies has higher accuracy.
TABLE 3 maximum correlation coefficient of the first 5 th order natural frequency ACE model
3) Uncertainty parameter quantization
Assuming that all the 9 uncertainty parameters obey normal distribution with a nominal value as a mean value and a coefficient of variation of 0.05-0.1, random sampling is performed according to the obeyed probability distribution function to obtain 18 sample values for simulating a field measured value, as shown in table 4.
TABLE 4 statistical eigenvalues of the parameters
In the actual engineering analysis, the measured values can be determined according to material experiments and data of the same engineering. The mean, standard deviation and their sample distribution characteristics of the uncertainty parameters were calculated using the proposed method from 18 raw samples of the 9 parameters measured in the field, as shown in table 5.
Statistical characterization of random sample mean, standard deviation of the 59 parameters in Table
As can be seen from table 5, the mean values of the sample mean values and the standard deviations are closer to the mean values and the standard deviations of the original samples, which means that the sample data obtained by random sampling effectively inherits the basic information of the original observation data. Meanwhile, it can be found from table 5 that the estimated mean and standard deviation have large variability, with a maximum coefficient of variation of 4%; the coefficient of variation of the sample standard deviation is larger than that of the sample mean, which shows that the mean and the standard deviation obtained by statistics of Bootstrap samples obtained by multiple times of sampling have certain uncertainty due to the random characteristic of random sampling.
4) Parametric sensitivity analysis
The results of the first 5 th order natural frequency sensitivity analysis are shown in fig. 4. As can be seen from FIG. 4, for the 1 st order natural frequency, the modulus of elasticity E of the main tower 2 And mass density M 2 The sensitivity is highest and is above 40%, and the rest parameters are insensitive parameters. Modulus of elasticity E of main beam 1 And mass density M 1 The influence degree on the natural frequency of the 2 nd order to the 5 th order is large; the second is the modulus of elasticity of the stay cable and the modulus of elasticity E of the main tower 2 And massDensity M 2 (ii) a Area A of cross section of inhaul cable 2 And A 3 Is sensitive to the 4 th and 5 th order natural frequencies, while A 1 It is more sensitive to the 3 rd order natural frequency. From table 2, it can be seen that the vibration characteristic of the 1 st order natural frequency is the main tower lateral bending, which is mainly controlled by the stiffness and mass of the main tower. The natural frequency of the 2 nd order to the 4 th order is vertical bending, is mainly controlled by the inhaul cable, the main beam and the inhaul cable together, and is consistent with the sensitivity analysis result. In order to verify the accuracy of the above analysis results, the sensitivity analysis was performed again using the analysis-by-variance sensitivity analysis method, and the results are shown in fig. 5. As can be seen from fig. 5: the results of the sensitivity analysis of the two methods are consistent.
5) Uncertainty analysis
The effect of the 9 uncertainty parameters on the first 5 natural frequencies is shown in table 7.
TABLE 7 first 5 order natural frequency uncertainty analysis results
From table 7, it can be seen: the first 5 natural frequencies calculated based on the deterministic model are all smaller than the measured values. The first 5 th order natural frequency means of the structure, which take into account the uncertainty of the parameters, are very close to the actual measurements. The standard deviation of the first 5 th order natural frequency is calculated to be in the range of 0.02-0.05. The value of the measured natural frequency is just in the calculated confidence interval range (the confidence level is 95%), so that the actual stress condition of the bridge can be more accurately reflected by adopting a structural dynamic response analysis result considering the uncertainty of the structural parameters.
Table 7 also shows the structural dynamic uncertainty analysis result based on Markov Chain Monte Carlo (MCMC), the relative error between the mean and the standard deviation estimated by the two methods is small, the maximum error of the mean is 0.0048, and the maximum relative error of the standard deviation is 0.28, which verifies the reliability of the calculation result of the method of this embodiment. It should be noted that the MCMC-based structure dynamic uncertainty quantification method needs to know the probability distribution and the distribution parameters obeyed by the uncertainty parameters in advance, and the present embodiment can quantify the structural response change caused by the parameter variation by using only a small amount of measured data, and is more suitable for the actual engineering analysis.
Table 8 shows the sampling characteristics of the first 5 th order natural frequency, and the mean value and the standard deviation of the first 5 th order natural frequency corresponding to each modified boottrap subsample have certain variability, and the variability of the standard deviation is larger than that of the mean value. Due to the inherent randomness of sampling, the mean value and the standard deviation of a sample with uncertainty parameters obtained based on limited data estimation have certain uncertainty, so that the distribution parameters of the structural dynamic response obtained by identification also have certain uncertainty. If these statistical uncertainties are not considered, the structural uncertainty analysis will have a certain influence.
TABLE 8 statistical characteristics of the mean and standard deviation of the first 5 th order natural frequency
In summary, in the present embodiment, information diffusion estimation is performed on the probability density function of each Bootstrap subsample, and a large number of improved Bootstrap subsamples are generated by adopting an acceptance-rejection method, so as to quantify uncertainty based on large sample data. Secondly, based on a response surface model, a change interval of the structural response is rapidly calculated, the influence degree of parameter uncertainty on the structural response is judged according to a defined interval sensitivity index, and the uncertainty of the response is quantified. The method can quantify the uncertainty of the civil engineering structure parameters under the conditions of unknown probability distribution and limited measured data through mathematical theory analysis and experiments, and calculate the influence of the parameter uncertainty on the structure response, and can be used for engineering structure damage diagnosis, health monitoring and structure analysis and optimization.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
The foregoing is directed to preferred embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow. However, any simple modification, equivalent change and modification of the above embodiments according to the technical essence of the present invention will still fall within the protection scope of the technical solution of the present invention.
Claims (3)
1. A bridge structure parameter uncertainty quantification and transfer method based on an improved self-service method is characterized by comprising the following steps: the method comprises the following steps:
step S1: improving Bootstrap sampling, and providing uncertainty quantification of bridge structure parameters based on an improved self-service method;
step S2: carrying out uncertainty transfer analysis on the physical parameters of the bridge structure based on the response surface model;
step S1 specifically includes the following steps:
step S11: acquiring a small amount of bridge structure parameter data measured in a field, wherein the bridge structure parameter data comprises the elastic modulus of a main tower, the mass density of the main tower, the elastic modulus of a main beam, the mass density of the main beam, the elastic modulus of a stay cable and the cross section area of the stay cable; x ═ X 1 ,x 2 ,...,x n ) Is a subsample from an unknown population F, θ (F) being an unknown parameter of the population; according to X ═ X 1 ,x 2 ,...,x n ) And constructing an empirical distribution function of the measured parameter sample, which is shown as the following formula:
in the formula, x (1) ≤x (2) ≤...≤x (n) Is x 1 ,x 2 ,...,x n Statistics after arrangement from small to large; n represents the capacity of the measured sample, and k represents a data arrangement serial number, wherein k is 1, 2.
Step S12: from F n (x) Repeatedly extracting the measured data of the bridge structure parameters in a release manner to form a Bootstrap subsample X * ＝(x 1 * ,x 2 * ,...,x n * ) (ii) a Diffusion according to informationTheory, probability density function f (x) to which Bootstrap subsamples are subjected * ) Comprises the following steps:
in the formula, mu (x) * ) Is a diffusion function defined as [ - ∞, + ∞ [ ]]A Borel measurable function; the constant delta n 'is the window width, and n' is the sub-sample capacity;
determining mu (x) using normal information diffusion estimation * ) Probability density function f (x) to which the uncertainty parameter of the bridge structure is estimated * ) Comprises the following steps:
in the formula (I), the compound is shown in the specification,
the window width is defined as the value of gamma is related to n', and sigma represents the standard deviation of the self-help sample;
step S13: according to an estimated probability density function f (x) * ) Generating pseudo random numbers by adopting an acceptance-rejection method, and selecting random number sequences of uniformly distributed sequences;
step S14: repeating the steps S12 to S13B times to obtain the statistic R of B improved Bootstrap subsamples j ** Thereby quantifying the uncertainty of the physical parameters of the bridge structure;
step S2 specifically includes the following steps:
step S21: determining test points by using physical parameters of the bridge structure as input and response of the bridge structure as output, and calculating dynamic response of the bridge corresponding to each test point by adopting a uniform design method; adopting an ACE algorithm to carry out nonparametric regression, and establishing an ACE response surface reflecting the relationship between the physical parameters of the bridge and the dynamic response of the bridge;
step S22: firstly, respectively connecting a bridge junction with uncertaintyB improved Bootstrap subsamples of the structural parameters are substituted into an ACE response surface, the average value of the remaining uncertain parameters is obtained, and the dynamic response of the bridge structure corresponding to the improved Bootstrap subsamples is rapidly calculated; secondly, respectively calculating confidence intervals of the improved Bootstrap subsamples and the corresponding responses thereof under the condition that the confidence degree is 95 percent to obtain B improved Bootstrap subsamples and corresponding response intervals, taking the average value of the B confidence intervals as the value intervals of the uncertainty parameters and the corresponding structural responses thereof, and substituting the value intervals into the following formula to obtain the sensitivity factor of each parameter
In the formula (I), the compound is shown in the specification,
representing model parameters x i The amount of change in the induced structural response y;
repeating the step S22, calculating the sensitivity factors of the rest bridge structure parameters with uncertainty, and judging the influence degree of the variation of each bridge structure parameter on the uncertainty of the structural dynamic response by comparing the sensitivity factors of each parameter;
step S23: substituting the B group of improved Bootstrap subsamples with high sensitivity factors into an ACE response surface, taking the average value of the uncertainty parameters with low sensitivity factors, calculating the B group of dynamic responses corresponding to the B group of improved Bootstrap subsamples, and further obtaining the statistical characteristic value of the B group of responses; the distribution of the B statistics is used for simulating the distribution of the dynamic response mean value and the standard deviation of the bridge structure, so that the sampling distribution and the distribution parameters of the statistical characteristic value of the bridge structure response probability are obtained, and the effects of multiple parameter variations on the structural dynamic response variation and the estimation error of the statistical characteristic of the structural response probability generated by random sampling are estimated.
2. The method for quantifying and transmitting the uncertainty of the parameters of the bridge structure based on the improved self-service method according to claim 1, wherein the method comprises the following steps: step S13 specifically includes: first, a certain distribution g (x) of a sample is selected as a suggested distribution, and then a constant M > 1 is determined, so that f (x) Mg (x) is established in the definition domain of x; then generating a suggested random number y subject to a probability density function g (x) and a random number U subject to a uniform distribution U (0, 1); finally, a probability function of the reception criterion is calculated
If u < h (y), receiving the generated random number y; otherwise, discarding the random number y, and generating the random number again; cycling the steps to obtain a plurality of improved Bootstrap subsamples X obeying the probability density function f (X) ** ＝(x 1 ** ,x 2 ** ,...,x m ** ) And m is more than n', calculating the statistic R of the improved Bootstrap subsample ** 。
3. The method for quantifying and transmitting the uncertainty of the bridge structure parameters based on the improved self-service method according to claim 1, wherein the method comprises the following steps: step S21 specifically includes: for structural response y and model parameters x 1 ,x 2 ,...,x i The multiple regression problem of (1) adopts an expression of an ACE regression model as follows:
where θ (y) is a transformation function of the structural response y; phi is a i (x i ) A transformation function of the model parameters; ε represents the fitting error; after the transformation function is obtained, the expression of the response y is expressed by the following formula, namely an optimal response surface model:
in the formula, Θ (·) represents an inverse function of θ.
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