Bridge reliability prediction method based on response surface methodology

Abstract

The present invention discloses a bridge reliability prediction method based on the response surface methodology, comprising the following steps: determining the statistical characteristics and probability distribution of random variables under the design state of the bridge structure, generating input sample points using the uniform design method; establishing a structural analysis model based on the bridge design data and operational conditions, calculating the target variables corresponding to the input samples, obtaining output samples, and forming a training sample with the input samples; normalizing the sample points, and constructing a response surface model based on an intelligent algorithm; standardizing the random variables, converting the constrained optimization problem into an unconstrained optimization problem using a penalty function, and obtaining the optimal weights of the random variables using an optimization algorithm; and establishing a mathematical model for solving the structural reliability index by constructing the prediction results of the response surface model. The present invention has the following beneficial effects: high calculation accuracy, fast estimation speed, and compatibility with existing finite element analysis software. It is suitable for reliability problems involving high structural analysis costs and highly nonlinear implicit performance functions.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge engineering, and in particular to a bridge reliability prediction method based on response surface methodology. Background Art

[0002] Bridge reliability research is currently a hot topic in civil engineering. In reliability analysis of complex bridge projects, the performance function is generally implicit, making it difficult to directly use algorithms such as the first-order second-moment method (FORM), the second-order second-moment method (SORM), and direct integration methods. While direct Monte Carlo simulation (MCS) is suitable for solving reliability problems with implicit performance functions and offers high computational accuracy, the MCS method requires a large number of samples to ensure accuracy. This is particularly true when the performance function values require finite element analysis. The enormous computational effort leads to extremely long computational times, significantly limiting the application of the MCS method in engineering. Therefore, using a small number of sampling points, the response surface of the implicit performance function is constructed using regression tools such as classical response surface methods (RSM), artificial neural networks (ANN), Kriging surrogate models, and support vector machines (SVM). This is then combined with conventional methods such as FORM, SORM, and MCS for reliability analysis. This approach effectively reduces the number of structural reanalysis attempts and significantly improves computational efficiency. It has become an important approach for reliability analysis of complex structures.

[0003] To address the low accuracy and precision of traditional response surface methods for bridge reliability calculations, a response surface method for bridge structure reliability prediction was proposed, combining the characteristics of a dynamic Bayesian network (DBN) and the PSOSA optimization algorithm. This method leverages the advantages of DBN in handling uncertainty and probabilistic reasoning, and the PSOSA algorithm's ability to better update particle swarm coordinates, thereby faster searching for optimal solutions. This method effectively improves the accuracy and efficiency of complex structural reliability calculations, overcomes the limitations of the classical response surface method for highly nonlinear structural reliability problems, and addresses the low computational efficiency of the MCS method and the excessive dependence of the calculation accuracy of existing response surface methods on the size and distribution of pre-set samples. Summary of the Invention

[0004] Given that the traditional response surface method has a highly nonlinear implicit functional function when solving bridge reliability problems, it leads to low fitting accuracy and difficulty in meeting precision requirements.

[0005] To solve the above problems, the present invention proposes a bridge reliability prediction method based on response surface methodology, which is characterized by comprising the following steps:

[0006] (1) Determine the statistical characteristics and probability distribution of random variables under the design state of the bridge structure, and use the uniform design method to generate input sample points:

[0007] Determine the statistical characteristics and probability distribution of random variables (design parameters) under the bridge structure design state based on the bridge design data, and use the uniform design method to generate sample points for the random variables;

[0008] The generation of random variable sample points adopts the uniform design method. According to the uniform design table, once the number of random variables and the number of sample point groups are determined, the sample point data can be automatically generated and can be directly consulted through the uniform design table;

[0009] (2) A structural analysis model is established based on the bridge design data and operating conditions. The target variables corresponding to each input sample are calculated to obtain the output sample, which is then combined with the input sample to form a training sample:

[0010] Based on the bridge structure design data and operational status, a structural analysis model is established using structural analysis software (ANSYS, ABAQUS, Midas, etc.). Based on the structural analysis model, the output results corresponding to each group of sample points are calculated, and the input data of the sample points and the corresponding calculation output data are used to form training samples;

[0011] (3) Normalize the sample points and construct a response surface model based on an intelligent algorithm;

[0012] (4) Normalize the random variables, use the penalty function to transform the constrained optimization problem into an unconstrained optimization problem, and use the optimization algorithm to obtain the optimal weight of the random variables;

[0013] (5) A mathematical model for solving structural reliability indicators is established based on the prediction results of the constructed response surface model.

[0014] Furthermore, in step (3), the sample points are normalized, and a basic DBN model is established based on the BN toolbox in MATLAB. The basic model is trained unsupervisedly and the model parameters are optimized by inputting the sample points to obtain a structure-related DBN response surface model.

[0015] Furthermore, step (3) performs normalization on the sample points. The training samples are normalized using a normalization method so that the normalized result is between 0 and 1. The normalization formula is:

[0016]

[0017] Where: x i is the sample point data, y i is the normalized result;

[0018] The normalized training samples were inserted into the DBN toolbox in MATLAB software to calculate the DBN response surface model for the DBN random variable. The response surface model was constructed by using the DBN toolbox in MATLAB software. The training sample data was input into the DBN toolbox algorithm to construct the response surface model.

[0019] Furthermore, in step (3), the sample points are normalized, and a basic DBN model is established based on the BN toolbox in MATLAB. Then, the basic model is trained unsupervised and the model parameters are optimized by inputting the sample points to obtain a structure-related DBN response surface model; wherein DBN can be expressed as (B0, B → ), where B0 is the static BN, which represents the probability distribution P(X0) of the node at the initial moment, B → It is a transition network containing two adjacent time slices, which represents the state transition probability between nodes in two adjacent time slices. The expression is:

[0020]

[0021] Where: is the i-th node in t time slices; The parent node of Can be used with In the same time slice, or in the previous time slice.

[0022] Furthermore, in step (4), the random variables are normalized. A penalty function is used to transform the constrained optimization problem into an unconstrained optimization problem. A fitness equation suitable for solving the PSOSA algorithm is constructed. The optimal positions of the search particles and particle swarms are updated through the PSOSA algorithm. The optimal weights of the random variables are iteratively obtained to support the unsupervised learning process of the DBN model. In the normalization of random variables, it is assumed that each random variable obeys a standard normal distribution. This process is called the normalization of random variables.

[0023] Furthermore, step (4) uses the penalty function method to transform the random variable constrained optimization problem into an unconstrained optimization problem. By introducing the function, the constrained optimization problem is transformed into an unconstrained optimization problem:

[0024]

[0025] Where F(x,σ) is the penalty function, f(x) is the objective function; σ is the penalty factor, It is a penalty term. The parameter x in F(x,σ) has no restrictions and can take any value.

[0026] The optimal weights of random variables are solved using the particle swarm optimization algorithm (PSOSA), and its principle is:

[0027]

[0028] Where: i is the particle number, d is the particle dimension number, k is the number of iterations, w is the inertia weight, c1 is the individual learning factor, c2 is the group learning factor, r1, r2 are random numbers in the interval [0-1] to increase the randomness of the search. is the velocity vector of particle i in the kth iteration with the dth dimension, is the position vector of particle i in the dth dimension in the kth iteration, is the historical optimal position of particle i in the d-th dimension in the k-th iteration, is the historical optimal position of the dth dimension in the kth iteration. By gradually iterating the above formula, the optimal weight of the random variable can be obtained.

[0029] Furthermore, step (5) establishes a mathematical model for solving the structural reliability index through the prediction results of the DBN model. In this process, it is necessary to update and optimize the samples of each DBN prediction model so that the DBN prediction model can well approximate the sample points until the model constructs a response surface with sufficient accuracy and can truly simulate the structural limit state function.

[0030] Furthermore, in step (5), the predicted results of the constructed response surface model are compared with the results of the true limit state function of the structure. When the predicted results of the response surface model converge to the results of the true limit state function of the structure, the response surface prediction results are directly used to calculate the structural reliability. When the predicted results of the response surface model do not converge to the true limit state function, it is necessary to update and optimize the samples of the DBN prediction model so that the DBN prediction model can well approximate the sample points until the model constructs a response surface with sufficient accuracy and can truly simulate the limit state function of the structure. The limit state function of the structure should be determined based on multiple factors such as the specific form of the structure and the reliability analysis object.

[0031] The beneficial effects of the present invention are:

[0032] 1) A hybrid rapid response surface method is proposed, combining a dynamic Bayesian network (DBN) with a particle swarm optimization algorithm (PSOSA) based on simulated annealing. This method leverages the advantages of DBN in handling uncertainty and probabilistic reasoning, while also taking advantage of the PSOSA algorithm's ability to better update particle swarm coordinates, leading to faster search for optimal solutions. This method effectively improves the accuracy and efficiency of reliability calculations for complex structures.

[0033] 2) It overcomes the limitations of the classical response surface method in solving highly nonlinear structural reliability problems, and solves the problems of low computational efficiency of the MCS method and excessive dependence of the existing response surface method's computational accuracy on the size and distribution of preset samples.

[0034] 3) Compared with traditional bridge reliability analysis methods, the DBN-PSOSA hybrid response surface method has the advantages of high calculation accuracy, fast estimation speed, and easy integration with existing finite element analysis software, which is convenient for engineering applications. It is especially suitable for reliability problems with high structural analysis costs and highly nonlinear implicit performance functions. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1a It is a flowchart of the present invention;

[0036] Figure 1b It is a specific flow chart of the present invention;

[0037] Figure 2 This is a schematic diagram of the typical DBN model structure;

[0038] Figure 3 is the PSOSA parameter optimization diagram;

[0039] Figure 4 This is the DBN prediction result graph. DETAILED DESCRIPTION

[0040] The following describes the specific implementation of the embodiment of the present invention in detail with reference to the accompanying drawings. It should be understood that the specific implementation described herein is only used to illustrate and explain the embodiment of the present invention and is not used to limit the embodiment of the present invention.

[0041] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0042] The present invention will be described in detail below with reference to the accompanying drawings and in conjunction with exemplary embodiments.

[0043] The present invention proposes a bridge reliability prediction method based on response surface methodology, which includes the following steps:

[0044] (1) Determine the statistical characteristics and probability distribution of random variables under the operational state of the bridge structure. Random variables include the geometric dimensions, material properties, boundary conditions, load conditions, etc. of the structure. Use the uniform design method to generate input sample points:

[0045] Random variables are determined based on the design data of the bridge structure, which contains information such as bridge geometric dimensions, material properties, boundary conditions, and load conditions.

[0046] The generation of random variable sample points adopts the uniform design method. According to the uniform design table, when the number of random variables and the number of sample point groups are determined, the sample point data can be automatically generated;

[0047] (2) Combined with the bridge design data and operation status, a structural analysis model is established using structural analysis software. Structural analysis software includes general finite element software such as ANSYS, ABAQUS, and professional finite element software such as Midas. Calculate the target variable corresponding to each input sample, obtain the output sample, and then form a training sample with the input sample:

[0048] Based on the bridge's structural design data and operational status, a structural analysis model was established using structural analysis software (ANSYS, ABAQUS, Midas, etc.). Based on the structural analysis model, the output results corresponding to each group of sample points were calculated. Training samples were formed using the input data of the sample points and the corresponding calculated output data.

[0049] Here, the input data of multiple groups of sample points and the corresponding calculation results are formed into a matrix to form a training sample;

[0050] (3) Normalize the sample points and build a basic DBN model based on the BN toolbox in MATLAB:

[0051] Step (3) normalizes the sample points. The training samples are normalized using the normalization method so that the normalized result is between 0 and 1. The normalization formula is:

[0052]

[0053] Where: x i is the sample point data, y i is the normalized result;

[0054] The normalized training samples were inserted into the DBN toolbox in MATLAB software to calculate the DBN response surface model for the DBN random variable. The response surface model was constructed by using the DBN toolbox in MATLAB software. The training sample data was input into the DBN toolbox algorithm to construct the response surface model.

[0055] DBN can be expressed as (B0,B → ), where B0 is the static BN, which represents the probability distribution P(X0) of the node at the initial moment, B → It is a transition network containing two adjacent time slices, which represents the state transition probability between nodes in two adjacent time slices. The expression is:

[0056]

[0057] Where: is the i-th node in t time slices; The parent node of Can be used with In the same time slice, or in the previous time slice, the typical DBN model structure is shown in Figure 2 As shown, Figure 2 (a) is the initial network; (b) is the transfer network; (c) is the DBN model with time slices. By inputting sample points to perform unsupervised training on the basic model and optimizing the model parameters, a structure-related DBN response surface model is obtained;

[0058] (4) The random variables are normalized and the penalty function is used to transform the constrained optimization problem into an unconstrained optimization problem. The fitness equation suitable for solving by the PSOSA algorithm is constructed, and the optimal positions of the search particles and particle swarms are updated by the PSOSA algorithm. The optimal weights of the random variables are iteratively obtained to support the unsupervised learning process of the DBN model. Figure 3 It is a state equation containing three random variables. The sample input points and sample output points are combined into training samples. The sample points are normalized and inserted into the basic DBN model for training. The parameters are optimized using the PSOSA algorithm. The optimization process of the optimal weight parameters w1-w3 of the model is as follows: Figure 3 shown.

[0059] Specifically, in the standard normalization of random variables, it is assumed that each random variable obeys the standard normal distribution. This process is called the standard normalization of random variables.

[0060] The penalty function method is used to transform the random variable constrained optimization problem into an unconstrained optimization problem. By introducing function (2), the constrained optimization problem is transformed into an unconstrained optimization problem:

[0061]

[0062] Where F(x,σ) is the penalty function, f(x) is the objective function; σ is the penalty factor, is a penalty term. The parameter x in F(x,σ) has no restrictions and can take any value, which transforms the constrained optimization problem into an unconstrained optimization problem.

[0063] The optimal weight of random variables is solved using the particle swarm optimization algorithm (PSOSA), and its formula is:

[0064]

[0065] Where: i is the particle number, d is the particle dimension number, k is the number of iterations, w is the inertia weight, c1 is the individual learning factor, c2 is the group learning factor, r1, r2 are random numbers in the interval [0-1] to increase the randomness of the search. is the velocity vector of particle i in the kth iteration with the dth dimension, is the position vector of particle i in the dth dimension in the kth iteration, is the historical optimal position of particle i in the d-th dimension in the k-th iteration, is the historical optimal position of the dth dimension in the kth iteration. By gradually iterating the above formula, the optimal weight of the random variable can be obtained;

[0066] Figure 3 In the embodiment, w1-w3 are three random variables. After 50 iterations of the PSOSA algorithm, the optimal weight calculation results of each random variable are 0.4571, 0.2362, and 0.3067, respectively. The optimal weights of the random variables are obtained by iteration to support the unsupervised learning process of the DBN model.

[0067] (5) The mathematical model for solving the structural reliability index is established through the prediction results of the DBN model. In this process, it is necessary to update and optimize the samples of each DBN prediction model so that the DBN prediction model can well approximate the sample points until the model constructs a response surface with sufficient accuracy and can truly simulate the structural limit state function. Figure 4 The DBN prediction results constructed using 30 sets of training samples generated by uniform design for random variables.

[0068] Specifically, step (5) compares the predicted results of the constructed response surface model with the results of the true limit state function of the structure. When the predicted results of the response surface model converge to the results of the true limit state function of the structure, the response surface prediction results are directly used to calculate the structural reliability. When the predicted results of the response surface model do not converge to the true limit state function, it is necessary to update and optimize the samples of the DBN prediction model so that the DBN prediction model can well approximate the sample points until the model constructs a response surface with sufficient accuracy and can truly simulate the limit state function of the structure.

[0069] The limit state function of the structure should be determined based on multiple factors such as the specific form of the structure and the analysis object of reliability. It is not convenient to directly give a specific function expression here.

[0070] Figure 4 It can be seen from the embodiment that the prediction results of the response surface model for the 30 groups of training samples in the random variable experimental design are consistent with the actual state function results of the structure, indicating that the DBN model can truly simulate the structural limit state function with good accuracy.

[0071] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise", "axial", "radial", "circumferential" and the like to indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as limiting the present invention.

[0072] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of the technical features being referred to. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present invention, "plurality" means at least two, such as two, three, etc., unless otherwise specifically defined.

[0073] In the present invention, unless otherwise specified or limited, the terms "installed," "connected," "connect," "fixed," etc. should be understood in a broad sense. For example, they can refer to fixed connection, detachable connection, or integration; mechanical connection, electrical connection, or communication; direct connection or indirect connection through an intermediate medium; internal communication between two elements or interaction between two elements, unless otherwise specified. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0074] In the present invention, unless otherwise expressly specified or limited, when a first feature is "above" or "below" a second feature, it may mean that the first and second features are in direct contact, or that the first and second features are in indirect contact through an intermediary. Furthermore, when a first feature is "above," "above," or "above" a second feature, it may mean that the first feature is directly above or diagonally above the second feature, or simply means that the first feature is at a higher level than the second feature. When a first feature is "below," "below," or "below" a second feature, it may mean that the first feature is directly below or diagonally below the second feature, or simply means that the first feature is at a lower level than the second feature.

[0075] In the description of this specification, the reference terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" mean that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art can combine and combine different embodiments or examples described in this specification and features of different embodiments or examples without contradiction.

[0076] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention.

Claims

1. The bridge reliability prediction method based on response surface methodology is characterized by: The following steps are involved: (1) Determine the statistical characteristics and probability distribution of random variables under the design state of the bridge structure and generate input sample points using the uniform design method; (2) Establish a structural analysis model based on bridge design data and operating conditions, calculate the target variables corresponding to each input sample, obtain output samples, and then form training samples with the input samples; (3) Normalize the sample points and construct a response surface model based on an intelligent algorithm; (4) Standard normalization of random variables, using penalty function to transform the constrained optimization problem into an unconstrained optimization problem, and using optimization algorithm to obtain the optimal weight of random variables; the standard normalization of random variables, using penalty function to transform the constrained optimization problem into an unconstrained optimization problem, constructing a fitness equation suitable for solving by PSOSA algorithm, and updating the optimal position of search particles and particle swarms by PSOSA algorithm, iteratively obtaining the optimal weight of random variables to support the unsupervised learning process of DBN model, wherein the standard normalization of random variables assumes that each random variable obeys standard normal distribution, and this process is the standard normalization of random variables; the penalty function method is used to transform the random variable constrained optimization problem into an unconstrained optimization problem, and by introducing function (3), the constrained optimization problem is transformed into an unconstrained optimization problem: Where F(x,σ) is the penalty function, f(x) is the objective function; σ is the penalty factor, is the penalty term. The parameter x in F(x,σ) has no restrictions and can take any value. (5) A mathematical model for solving structural reliability indicators is established based on the prediction results of the constructed response surface model.

2. The bridge reliability prediction method based on response surface methodology according to claim 1, characterized in that: In step (3), the sample points are normalized, and a basic DBN model is established based on the BN toolbox in MATLAB. The basic model is trained unsupervised by inputting sample points and the model parameters are optimized to obtain a structure-related DBN response surface model.

3. The bridge reliability prediction method based on response surface methodology according to claim 2, characterized in that: Step (3) normalizes the sample points. The training samples are normalized using the normalization method so that the normalized result is between 0 and 1. The normalization formula is: Where: x i is the sample point data, y i is the normalized result; The normalized training samples are substituted into the DBN toolbox in MATLAB software to calculate the DBN response surface model of the DBN random variable; the response surface model is constructed by using the DBN toolbox in MATLAB software. The training sample data is input into the algorithm of the DBN toolbox to construct the response surface model.

4. The bridge reliability prediction method based on response surface methodology according to claim 3, characterized in that: Step (3) normalizes the sample points and establishes a basic DBN model based on the BN toolbox in MATLAB. Then, the basic model is trained unsupervised and the model parameters are optimized by inputting the sample points to obtain a structure-related DBN response surface model. DBN can be expressed as (B0, B → ), where B0 is the static BN, which represents the probability distribution P(X0) of the node at the initial moment, B → It is a transition network containing two adjacent time slices, which represents the state transition probability between nodes in two adjacent time slices. The expression is: Where: is the i-th node in t time slices; The parent node Can be used with In the same time slice, or in the previous time slice.

5. The bridge reliability prediction method based on response surface methodology according to claim 1, wherein: In step (5), a mathematical model for solving the structural reliability index is established through the prediction results of the DBN model. In this process, it is necessary to update and optimize the samples of each DBN prediction model so that the DBN prediction model can well approximate the sample points until the model constructs a response surface with sufficient accuracy and can truly simulate the structural limit state function.

6. The bridge reliability prediction method based on response surface methodology according to claim 5, characterized in that: Step (5) compares the prediction results of the constructed response surface model with the results of the real limit state function of the structure. When the prediction results of the response surface model converge to the results of the real limit state function of the structure, the response surface prediction results are directly used to calculate the structural reliability; when the prediction results of the response surface model do not converge to the real limit state function, it is necessary to update and optimize the samples of the DBN prediction model so that the DBN prediction model can well approximate the sample points until the model constructs a response surface with sufficient accuracy and can truly simulate the structural limit state function.

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