Improved random response surface method for identification of uncertain parameters

Abstract

The application discloses an improved random response surface uncertain parameter identification method, which comprises the following steps: converting structure uncertain parameters p i into a function of standard random variables ξ i , expanding structure response y by using a chaos polynomial, and solving chaos polynomial coefficients a j ; through orthogonal test design of statistical characteristic values of the uncertain parameters p=[p1, p2, …], the chaos polynomial coefficients a j are expanded into a quadratic response surface form about the statistical characteristic values of the structure uncertain parameters pi, and a random response surface model of structure positive problem calculation is constructed. The application has the beneficial effects that the application proposes an uncertain parameter identification method of the structure random response surface by considering the structure uncertain parameters, constructing the structure random response surface by using the chaos polynomial and the quadratic response surface method, establishing a structure parameter identification inverse problem objective function, and identifying the statistical characteristic values of the structure uncertain parameters in steps, so that the calculation efficiency is higher than that of a traditional random response surface optimization method, and the application has very important engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of uncertain parameter identification and uncertain model correction technology, and particularly relates to an improved method for identifying uncertain parameters of stochastic response surfaces. Background Technology

[0002] An accurate numerical or finite element model is a prerequisite for reliability analysis and uncertainty optimization design of engineering structures. A common practice in engineering structural analysis is to correct the structural finite element model through parameter identification. This is a comprehensive technique involving initial finite element modeling, dynamic testing, and numerical methods. Currently, most model correction and parameter identification methods are deterministic, which significantly limits the effective application of model correction techniques in actual structures. However, real-world engineering structures often contain a certain degree of uncertainty. Therefore, considering structural parameters as uncertain parameters helps improve the reliability of analysis results and provides a scientific basis for decision-making.

[0003] Methods for model modification and parameter identification considering uncertainty are a deepening and extension of the theory of deterministic model modification and parameter identification in a statistical sense. They involve theoretical methods such as probability statistics and fuzzy sets, which greatly increases the complexity of the problem and the difficulty of model modification. Currently, known research results both domestically and internationally are very limited, and relevant theoretical research is urgently needed. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide an improved method for identifying uncertain parameters of a stochastic response surface.

[0005] This improved method for identifying uncertain parameters in a stochastic response surface includes the following steps:

[0006] S1: The structural uncertainty parameter p i Transform into a standard random variable ξ i The function is used to expand the structural response y using chaotic polynomials, and the chaotic polynomial coefficients a are solved. j ;

[0007] S2: Through orthogonal experimental design using the statistical eigenvalues ​​of the uncertain parameter p = [p1, p2, ...], the chaotic polynomial coefficients a j Expanding to the structural uncertainty parameter p i The quadratic response surface form of statistical eigenvalues ​​is derived, and a stochastic response surface model for calculating the structural positive problem is constructed.

[0008] S3: Construct an inverse problem objective function using the error between the random response surface model and the measured response statistical characteristic values, and identify the statistical characteristic values ​​of the uncertain parameters step by step.

[0009] As a preferred option, step one specifically involves:

[0010] S1.1: The structural uncertainty parameter p i Transform into a standard normal random variable ξ i The function is shown in equation (1).

[0011] F(p i )=Φ(ξ i →p i =F -1 [Φ(ξ i )]#(1)

[0012] Where, p i Let ξ be the i-th structural uncertainty parameter of a known distribution. i For p i The corresponding standard normal random variable; F(p) i ) and Φ(ξ i p respectively i and ξ i The probability distribution function, F -1 It is the inverse function of F;

[0013] S1.2: Expand the structural response y into a chaotic polynomial with a Hermite basis, as shown in equation (2).

[0014]

[0015] Where y is the structural response, q is the truncation order of the chaotic polynomial, and a j and H j Let be the coefficient of the j-th term in the chaotic polynomial and the Hermite basis function, respectively, and let ξ be the set of standard normal random variables [ξ1, ξ2, ...] corresponding to the structural uncertainty parameters [p1, p2, ...].

[0016] S1.3: Using the probability collocation method to analyze the standard normal random variable ξ i Sampling is performed, and the collocational samples of the q-th order chaotic polynomial are the roots of the (q+1)-th order Hermite polynomial. The response of the structure under the collocational samples is calculated, and the coefficients a of the chaotic polynomial are obtained by solving using the least squares method. j .

[0017] As a preferred option, step two specifically involves:

[0018] S2.1: Orthogonal experimental design is performed on the statistical eigenvalues ​​of the structural uncertainty parameter p = [p1, p2, ...], and the n chaotic polynomial coefficients a under the n experimental designs are calculated according to equation (2). j a is obtained by using the least squares method j Expanding to the structural uncertainty parameter p iThe quadratic response surface form of the statistical eigenvalues ​​is shown in equation (3).

[0019]

[0020] Where β j Let a be the coefficient of the j-th polynomial. j The corresponding quadratic response surface coefficients, x k and x l Let p be the kth and 1st statistical eigenvalues ​​corresponding to the structural uncertainty parameter p, and m be the number of statistical eigenvalues. Equation (3) can be written in reduced form as follows:

[0021] a j =f j (x)#(4)

[0022] Among them, f j The polynomial coefficients a j The corresponding quadratic response surface function, where x is the statistical eigenvalue [x1, x2, ..., x] corresponding to the structural uncertainty parameter p. m ];

[0023] S2.2: Combining equations (2) and (4), we obtain the stochastic response surface model of the statistical characteristic value x corresponding to the structural uncertainty parameter p, as shown in equation (5).

[0024]

[0025] As a preferred option, step S3 specifically involves:

[0026] S3.1: Constructing the mean error function of the structural response As shown in equation (6)

[0027]

[0028] in, Let be the mean of the t-th response of the structural random response surface model. Given the mean of the t-th experimental response of the structure, and keeping the standard deviation of the initial statistical characteristic value of the structural uncertainty parameter unchanged, construct the optimization inverse problem according to the objective function described in Equation (6), and perform several iterations in combination with the single objective optimization algorithm until convergence, and identify the mean of the structural uncertainty parameter;

[0029] S3.2: Constructing the standard deviation error function of structural response As shown in equation (7)

[0030]

[0031] in, Let be the standard deviation of the t-th response of the structural stochastic response surface model. Let p be the standard deviation of the t-th test response of the structure. Keep the mean of the modified structural uncertainty parameter unchanged. Based on the objective function described in equation (7), construct an optimization inverse problem and combine it with a single-objective optimization algorithm to perform several iterations until convergence, and identify the standard deviation of the structural uncertainty parameter p.

[0032] The beneficial effects of this invention are:

[0033] This invention discloses an improved method for identifying uncertain parameters in stochastic response surfaces. It proposes a method for identifying uncertain parameters in structural stochastic response surfaces by considering structural uncertain parameters and constructing the method using chaotic polynomials and quadratic response surfaces. By establishing an objective function for the inverse problem of structural parameter identification, the method identifies the statistical characteristic values ​​of structural uncertain parameters step by step. The computational efficiency is higher than that of traditional stochastic response surface optimization methods, and it has significant engineering application value. Attached Figure Description

[0034] Figure 1 This is a flowchart of the method of the present invention;

[0035] Figure 2 This is a geometric schematic diagram of a simply supported beam structure in a specific embodiment of the present invention;

[0036] Figure 3 The objective function for the structural response mean Iterative process diagram;

[0037] Figure 4 The objective function is the standard deviation of the structural response. Iterative process diagram;

[0038] Figure 5 Error map for identifying statistical characteristic values ​​of structural uncertainty parameters. Detailed Implementation

[0039] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

[0040] As one example, such as Figure 1 As shown, an improved method for identifying uncertain parameters of a stochastic response surface is proposed, for example... Figure 2 Modal analysis was performed on the simply supported beam shown. The total length of the simply supported beam structure is 3m, the cross-sectional dimensions are b×b=0.25m×0.25m, and the structural density is 2400kg / m³. 3 The elastic modulus E = 30 GPa and the Poisson's ratio is 0.2. The elastic modulus E and the cross-sectional dimension b are structural uncertainty parameters that follow a normal distribution. After normalization to nominal values, the distribution characteristics of the two parameters are N...E (1, 0.1) 2 ), N b (1, 0.2) 2 The improved method for identifying uncertain parameters of a stochastic response surface according to the present invention includes the following steps:

[0041] S1: Transform the structural uncertainty parameters into a function of standard random variables, and expand the structural response using chaotic polynomials;

[0042] S1.1: The uncertainty parameter p of the structural normal distribution i Transform into a standard normal random variable ξ i The function is shown in equation (1):

[0043] F(p i )=Φ(ξ i →p i =F -1 [Φ(ξ i )]=μ i +σ i ξ i (1)

[0044] Where F and Φ are the i-th parameter p of the structure, respectively. i (where p1 = E, p2 = b) and the corresponding standard normal random variable ξ i The corresponding probability distribution functions, F -1 μ is the inverse function of F; i and σ i These are the mean and standard deviation of a normally distributed random variable after standardization.

[0045] S1.2: Expand the modal frequencies y of each order of the structure into chaotic polynomials with Hermite basis, as shown in equation (2):

[0046]

[0047] Where y represents the modal frequencies of the structure, q represents the truncation order of the chaotic polynomial, and in the example q = 3, a j and H j Let be the coefficient of the j-th term in the chaotic polynomial and the Hermite basis function, respectively, and let ξ be the set of standard normal random variables [ξ1, ξ2, ...] corresponding to the uncertain parameters [p1, p2, ...].

[0048] S1.3: Using the probability collocation method to analyze the standard normal random variable ξ i Sampling is performed, and the collocational samples of the q-th order chaotic polynomial are the roots of the (q+1)-th order Hermite polynomial. The response of the structure under the collocational samples is calculated, and the coefficients a of the chaotic polynomial are solved using the least squares method. j.

[0049] S2: Obtain the quadratic response surface of chaotic polynomial coefficients through orthogonal experimental design of the statistical eigenvalues ​​of uncertain parameters, and construct a stochastic response surface model for structural forward problem calculation;

[0050] S2.1: Orthogonal experimental design is performed on the statistical characteristic values ​​[μ1, σ1, μ2, Φ2] of the structural uncertainty parameter p = [p1, p2]. The n chaotic polynomial coefficients a under the n experimental designs are calculated according to equation (2). j a is obtained by using the least squares method j Expanding to the structural uncertainty parameter p i The quadratic response surface form of the statistical eigenvalues ​​is shown in equation (3):

[0051]

[0052] Where, β j Let a be the coefficient of the j-th polynomial. j The corresponding quadratic response surface coefficients, x k and x l Let p be the kth and lth statistical eigenvalues ​​corresponding to the structural uncertainty parameter p. Equation (3) can be written in reduced form as follows:

[0053] a j =f j (x) (4)

[0054] Among them, f j The polynomial coefficients a j The corresponding quadratic response surface function, where x is the statistical characteristic value [μ1, σ1, μ2, σ2] corresponding to the structural uncertainty parameter p;

[0055] S2.2: Combining equations (2) and (4), we obtain the stochastic response surface model with respect to the statistical eigenvalue x of the structural uncertainty parameter p, as shown in equation (5):

[0056]

[0057] S3: Construct an inverse problem objective function using the error between the random response surface model and the measured response statistical characteristic values, and identify the statistical characteristic values ​​of the uncertain parameters step by step.

[0058] S3.1: Constructing the mean error function of the structural response As shown in equation (6):

[0059]

[0060] in, Let be the mean frequency of the t-th modal of the structural random response surface model. Let N be the experimental mean of the t-th modal frequency of the structure, and let N be the initial statistical characteristic values ​​of the structural uncertainty parameters. E (0.8, 0), N b (0.8, 0), keeping the standard deviation constant, select the fourth-order modal frequency of the structure, construct the optimization inverse problem according to the objective function described in equation (6), and perform multiple iterations using a single-objective optimization algorithm, such as Figure 3 As shown, the structural response mean error function The second convergence was achieved, identifying the mean of the uncertain parameters of the structure;

[0061] S3.2: Constructing the standard deviation error function of structural response As shown in equation (7):

[0062]

[0063] in, Let be the standard deviation of the t-th bending mode frequency of the structural random response surface model. The standard deviation of the t-th bending modal frequency of the structure is used. To maintain the mean of the corrected structural uncertainty parameters, the fourth modal frequency of the structure is selected. An inverse optimization problem is constructed based on the objective function described in equation (7), and multiple iterations are performed using a single-objective optimization algorithm. Figure 4 As shown, the structural response standard deviation error function The second convergence was achieved, and the standard deviation of the structural uncertainty parameter was identified.

[0064] like Figure 5 As shown, the absolute error of the statistical characteristic values ​​of structural uncertain parameters obtained by the improved random response surface uncertain parameter identification method proposed in this invention is significantly reduced compared with the initial value. It can be seen that this invention can reduce the identification error of the statistical characteristic values ​​of structural uncertain parameters and has higher computational efficiency than the traditional random response surface optimization method.

Claims

1. An improved method for identifying uncertain parameters of a stochastic response surface, characterized in that, Includes the following steps: S1: Structural uncertainty parameters Transform into a standard normal random variable The function is used to expand the structural response y using chaotic polynomials, and the coefficients of the chaotic polynomials are solved. ; S2: Through structural uncertainty parameters Orthogonal experimental design of statistical eigenvalues, using chaotic polynomial coefficients Expanding to structural uncertainty parameters The quadratic response surface form of statistical eigenvalues ​​is derived, and a stochastic response surface model for calculating the structural positive problem is constructed. S3: Construct an inverse problem objective function using the error between the random response surface model and the measured response statistical characteristic values, and identify the statistical characteristic values ​​of the uncertain parameters step by step; Step S1 is as follows: S1.1: Structural uncertainty parameters Transform into a standard normal random variable ξ i The function is shown in equation (1). （1） in, Let i be the structural uncertainty parameter of a known distribution. for The corresponding standard normal random variable; and respectively and The probability distribution function, for The inverse function; S1.2: Expand the structural response y into a chaotic polynomial with a Hermite basis, as shown in equation (2). （2） Where y is the structural response and q is the truncation order of the chaotic polynomial. and Let be the coefficient of the j-th term in the chaotic polynomial and the Hermite basis function, respectively, and ξ be the structural uncertainty parameter. The corresponding standard normal random variable set [ξ1,ξ2,…]; S1.3: Using the probability matching method for standard normal random variables Sampling is performed, and the collocational samples of the q-th order chaotic polynomial are the roots of the (q+1)-th order Hermite polynomial. The response of the structure under the collocational samples is calculated, and the coefficients of the chaotic polynomial are obtained by solving using the least squares method. ; Step S2 is as follows: S2.1: For structural uncertainties Orthogonal experimental design is performed using statistical eigenvalues, and the coefficients of n chaotic polynomials under n experimental designs are calculated according to equation (2). By using the least squares method Expanding to structural uncertainty parameters The quadratic response surface form of the statistical eigenvalues ​​is shown in equation (3). （3） in, and Structural uncertainty parameters The corresponding k-th and l-th statistical characteristic values, where m is the number of statistical characteristic values, can be written in reduced form as follows: （4） in, Polynomial coefficients The corresponding quadratic response surface function, where x is the structural uncertainty parameter. The corresponding statistical characteristic values ​​[x1, x2, ..., x] m ]; S2.2: Combining equations (2) and (4), we obtain the structural uncertainty parameters. The stochastic response surface model for the corresponding statistical characteristic value x is shown in equation (5). （5）。 2. The improved method for identifying uncertain parameters of a stochastic response surface according to claim 1, characterized in that, Step S3 is as follows: S3.1: Constructing the mean error function of the structural response As shown in equation (6) （6） in, Let be the mean of the t-th response of the structural random response surface model. Given the mean of the t-th experimental response of the structure, and keeping the initial statistical characteristic values ​​of the structural uncertainty parameters constant at their standard deviations, construct an inverse optimization problem according to the objective function described in equation (6), and perform several iterations using a single-objective optimization algorithm until convergence, thereby identifying the structural uncertainty parameters. Mean; S3.2: Constructing the standard deviation error function of structural response As shown in equation (7) （7） in, Let be the standard deviation of the t-th response of the structural stochastic response surface model. Let be the standard deviation of the structural response at the t-th test, and keep the corrected structural uncertainty parameters. With the mean unchanged, an inverse optimization problem is constructed based on the objective function described in equation (7). This problem is then iterated several times using a single-objective optimization algorithm until convergence is achieved, and structurally uncertain parameters are identified. The standard deviation.

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