A method for predicting dynamics response and reliability of a hybrid uncertain system

Abstract

This invention discloses a method for predicting the dynamic response and reliability of a hybrid uncertain system, belonging to the field of uncertain system dynamics and reliability prediction. The implementation method of this invention is as follows: A polynomial surrogate model of the hybrid uncertain system is represented based on the PCLM method. An orthogonal polynomial basis corresponding to the random variables is established based on data-driven methods: a one-dimensional orthogonal polynomial basis is represented based on the origin moments of the random variables; a tensor product operation is performed to obtain a multi-dimensional orthogonal polynomial basis corresponding to the random variables; and Legendre polynomials are used as the orthogonal polynomial basis corresponding to the interval parameters. The coefficient matrix γ in the polynomial surrogate model of the hybrid uncertain system is solved using a double-weighted least squares method. Based on the established polynomial surrogate model, the dynamics and reliability of the hybrid uncertain system are predicted respectively. This invention can improve the accuracy and efficiency of predicting the dynamic response and reliability of hybrid uncertain systems.

Description

Technical Field

[0001] This invention relates to a method for predicting the dynamic response and reliability of a system with mixed uncertainties, and more particularly to a method for approximating the response and performance functions of a system with mixed uncertainties based on a weighted data-driven polynomial surrogate model, belonging to the field of dynamics and reliability prediction of uncertain systems.

[0002] Background Method

[0003] In most engineering problems, the material properties, dynamic parameters, loads, boundary conditions, and initial conditions of a system are generally uncertain, and the accuracy of model parameters directly affects the prediction of the system's dynamic behavior and the precision of control. Therefore, considering the impact of uncertainty on system dynamics and reliability is crucial. Currently, most research categorizes uncertain parameters into probabilistic and non-probabilistic uncertainties. By adjusting probabilistic and non-probabilistic methods, the uncertainties of different types of parameters can be represented more effectively. For uncertain parameters with sufficient information, they can be treated as random variables using a predefined probability distribution function. For situations with insufficient information, such as when only the parameter boundaries are known, interval variables are generally used to describe this type of uncertain parameter.

[0004] Assessing the impact of uncertain parameters on the dynamic response of a system has gradually become a fundamental means and important method for quantifying uncertainties in a system. With the increasing complexity of engineering system design, uncertainty propagation faces numerous challenges such as "low accuracy" and the "curse of dimensionality." For dynamic models with both stochastic and interval input parameters, the polynomial-chaos-Chebyshev-interval (PCCI) method proposed by Wu et al. has higher computational efficiency than nested scan methods and Monte Carlo simulations. Since the Legendre interval model based on the scan method has higher accuracy and efficiency than the Chebyshev inclusion function, Feng proposed the polynomial-chaos-Legendre-metamodel (PCLM) method to solve mixed uncertain structural models. Compared to the PCCI method, the PCLM method can obtain more accurate response results.

[0005] Furthermore, the purpose of reliability analysis for engineering systems is to assess the failure probability of engineering systems with uncertainties. For engineering systems with mixed uncertainties, unlike reliability assessments containing only random uncertainties, the failure probability in structural reliability analysis, which involves both random and interval variables, is an interval value. Traditional Monte Carlo simulation (MCS) can obtain the maximum and minimum failure probabilities of structural reliability relatively accurately; however, when the engineering system model is complex, this method becomes computationally inefficient. The kriging-assisted sampling method for structural reliability analysis with mixed uncertainties proposed by Xiao et al., and the reliability analysis method combining polynomial-chaotic kriging and adaptive radial importance sampling proposed by Pan et al., can effectively improve the computational efficiency of reliability analysis.

[0006] The mixed uncertain systems studied above all contain interval parameters and random parameters, with the random parameters following a normal distribution. For non-normally distributed random variables, researchers typically use Rackwitz-Fiessler transforms, Rosenblatt methods, etc., to transform them into independent normally distributed random variables. However, the strong nonlinearity introduced during the transformation process leads to a decrease in computational accuracy. Furthermore, when the random parameters in a mixed uncertain system are defined by a dataset with an unknown distribution, the dynamic response and reliability analysis of the mixed uncertain system employ traditional reference methods. The reference methods for calculating the dynamic response of mixed uncertain systems are nested sweep methods and Monte Carlo simulations (SMMCS), while the reference method for reliability analysis is MCS. As reference methods, SMMCS and MCS are inefficient and difficult to apply in practical engineering fields.

[0007] Therefore, improving the accuracy of dynamic response and reliability prediction in mixed uncertainty systems containing non-normally distributed random variables is a research area that urgently needs breakthroughs at this stage. Simultaneously, improving the efficiency of dynamic response and reliability prediction in mixed uncertainty systems for random parameters defined by datasets with unknown distributions, or random variables with discrete probability densities, is also a research area that urgently needs breakthroughs at this stage. Summary of the Invention

[0008] This invention addresses the problem of predicting the dynamic response and reliability of mixed uncertain systems in engineering. The method for predicting the dynamic response and reliability of mixed uncertain systems aims to solve the following problem: When establishing a polynomial surrogate model for a mixed uncertain system containing interval parameters and random parameters, the orthogonal polynomial basis for the interval parameters is based on Legendre polynomials, while the orthogonal polynomial basis for the random parameters is no longer a fixed type of polynomial but is established using a data-driven approach. Based on this, the dynamic response or reliability of the mixed uncertain system is predicted. Therefore, for mixed uncertain systems containing interval parameters and non-normally distributed random parameters, this invention can improve the accuracy of dynamic response and reliability prediction; simultaneously, for mixed uncertain systems containing interval parameters and random parameters defined by an unknown distribution of data sets, this invention can improve the efficiency of dynamic response and reliability prediction. Since this invention only considers the statistical moment information of random variables when establishing a polynomial surrogate model, and is independent of the distribution type of random parameters, it can be applied to engineering systems with interval parameters and random parameters with arbitrary distributions (including continuous, discrete, and unknown distributions) in the engineering field. It establishes a weighted, data-driven polynomial surrogate model for mixed uncertain systems, improves the approximation accuracy of the polynomial surrogate model for the original mixed uncertain system, and solves the problem of predicting the dynamic response and reliability of related mixed uncertain systems in the engineering field.

[0009] This invention is achieved through the following method:

[0010] This invention discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system: A polynomial surrogate model of the hybrid uncertainty system is represented based on the PCLM method. An orthogonal polynomial basis for the random variables is established in a data-driven manner: a one-dimensional orthogonal polynomial basis is represented based on the origin moments of the random variables; then, a tensor product operation is performed on the one-dimensional orthogonal polynomial basis of each dimension of the random variables to obtain a multi-dimensional orthogonal polynomial basis for the random variables. Legendre polynomials are used as the orthogonal polynomial basis for the interval parameters. The coefficient matrix γ in the polynomial surrogate model of the hybrid uncertainty system is solved using a double-weighted least squares method. Based on the established polynomial surrogate model, the dynamics and reliability of the hybrid uncertainty system are predicted.

[0011] This invention discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system, comprising the following steps:

[0012] Step 1: Represent the polynomial surrogate model of the hybrid uncertainty system based on the PCLM (polynomial-chaos-Legendre-metamodel) method.

[0013] The output response y = F(ξ, [η]) contains an n-dimensional random variable ξ = (ξ1, ξ2, ..., ξn). n ) and m-dimensional interval variables [η] = ([η1], [η2], ... [η] m ]), and establish a polynomial proxy model for the response y.

[0014] The polynomial proxy model established in step 1 is shown in formula (1):

[0015]

[0016] Where, γ i A function of the interval parameter [η] is expressed as:

[0017]

[0018] in, It is a one-dimensional orthogonal polynomial basis. orthogonal polynomial basis The order of , satisfying Ξ i (ξ) is a base of one-dimensional orthogonal polynomials. An n-dimensional orthogonal polynomial of order no more than p is constructed through tensor product operations. In equation (2), Ψ j ([η])(j=0,...,T-1) is the Legendre polynomial, γ i,j Let I represent the elements of the coefficient matrix γ, and let I = (p+n)! / (p!n!), J = (q+m)! / (q!m!), where q is the order of the expansion of the orthogonal polynomial corresponding to the interval variable. The following establishes a one-dimensional orthogonal polynomial basis for the random variable.

[0019] Step 2: Establish the orthogonal polynomial basis for the random variables based on data-driven methods, and use Legendre polynomials as the orthogonal polynomial basis for the interval parameters.

[0020] Step 2.1: Represent a one-dimensional orthogonal polynomial basis based on the origin moments of the random variables.

[0021] First, consider the r-th dimension random variable ξ. r The corresponding one-dimensional orthogonal polynomial basis The definition is as follows:

[0022]

[0023] in, To find the coefficients of the one-dimensional orthogonal polynomial, k rLet r be the order of the one-dimensional orthogonal polynomial. For simplicity, the dimension identifier r in formula (3) is omitted below, so formula (3) is abbreviated as:

[0024]

[0025] When ξ is a continuous random variable, its k-th order raw moment is:

[0026] μ k =∫ ξ∈Ω ξ k dΓ(ξ),k=0,...,p (5)

[0027] Since obtaining orthogonal bases depends on the moments of the input samples, we can consider continuous or discrete probability density functions, or even datasets with unknown distributions. If ξ is a discrete random variable, then its k-th raw moment is:

[0028]

[0029] If the random variable ξ is represented by only one set of sampling points {ξ1,ξ2,...,ξ} Q}, then its k-th order raw moment is approximately expressed as:

[0030]

[0031] Statistical moments are quantitative measures describing the shape of a set of random samples or a probability distribution. The 0th-order raw moment represents the integral of the probability density function, and by definition, it is always 1; the first moment represents the mean, the second moment represents the variance, the third moment represents the skewness, the fourth moment represents the kurtosis, and so on. It is important to note that insufficient sample data can introduce some error. Based on the information from the raw moments, we obtain:

[0032]

[0033] For the coefficients in formula (6) The matrix is ​​obtained by inverting the matrix on the left side of equation (6). However, as the order k increases, this matrix may become very ill-conditioned. Therefore, an alternative method for matrix operations on the Hankel moment matrix is ​​considered. The Hankel moment matrix is ​​defined as:

[0034]

[0035] If a random variable is defined by a sampled dataset, then the set of samples must be determined in the Hamburger sense, which means that the Hankel moment matrix is ​​a positive definite matrix, i.e., det(H) > 0. The matrix obtained by performing Cholesky decomposition on the above Hankel moment matrix is:

[0036]

[0037] Make H = R T R. Inverting matrix R yields...

[0038]

[0039] According to Mysovskih theory, matrix R -1 The elements in the formula (4) represent the orthogonal basis, that is:

[0040]

[0041] To avoid inverting the matrix, a one-dimensional orthogonal basis is obtained using a three-term recursive formula:

[0042] ξχ (k-1) (ξ)=b k-1 χ (k-2) (ξ)+a k χ (k-1) (ξ)+b k χ (k) (ξ) (11)

[0043] The coefficient 'a' of the three recursive terms k and b k satisfy:

[0044]

[0045] Where R 0,0 =1,R 0,1 =0. Therefore, we can see that the coefficients in the three recursive formulas are all elements of matrix R, thus avoiding the need to invert matrix R.

[0046] Step 2.2: Perform tensor product operation on the one-dimensional orthogonal polynomial basis of each dimension of the random variable to obtain the multidimensional orthogonal polynomial basis corresponding to the random variable.

[0047] For a multidimensional input system, the tensor product operation is performed on the one-dimensional orthogonal polynomials corresponding to each dimension, thus constructing a multidimensional orthogonal polynomial:

[0048]

[0049]

[0050] in It is a multidimensional index, containing information on all possible combinations of products for each univariate basis function. Furthermore, since ξ1, ξ2, ..., ξ nThese are the original random variables. The order of magnitude of the samples for each dimension of the random variable may differ significantly, leading to an "ill-conditioned" matrix that affects computational accuracy and efficiency. Therefore, it is necessary to normalize the random variables for each dimension, that is, to normalize each dimension ξ. i (i=1,...,n), there are:

[0051]

[0052] At this time ξ′ i The mean is 0 and the standard deviation is 1. Then, the orthogonal polynomial basis for each dimension is constructed using steps 2.1-2. Finally, the multidimensional orthogonal polynomial basis is obtained through tensor product operation.

[0053] Step 2.3: Use Legendre polynomials as the orthogonal polynomial base corresponding to the interval parameters.

[0054] Step 3: Solve for the coefficient matrix γ in the polynomial surrogate model of the mixed uncertainty system using the double-weighted least squares method.

[0055] The coefficient matrix is ​​solved using two least squares regressions. Since the selection of the sampling technique is crucial to the robustness of the least squares regression method, consistent coherent sampling is used for samples with interval parameters; for samples with random parameters, the optimal sampling points and weights can be calculated based on three recursive relationships. The specific implementation method is as follows:

[0056] First, construct a symmetric tridiagonal Jacobi matrix:

[0057]

[0058] The elements of this matrix are obtained through a three-term recurrence relation, and matrix J is positive definite. The eigenvalues ​​of matrix J are the roots corresponding to the p-th order polynomial, and the weight of each root is:

[0059]

[0060] Among them, v 1,i Let represent the first component of the standard eigenvector corresponding to the i-th eigenvalue. Then, a sampling method similar to sparse grid numerical integration is used to obtain samples of random parameters and the weights corresponding to each sample group. The 1D integration points are obtained from the eigenvalues ​​and eigenvectors of equation (16). and weight Then, a tensor product operation is performed on this 1D integral to obtain an n-dimensional sparse grid of points with l-level precision (l≥0):

[0061]

[0062] Where i1,...,i nFor the multiple exponents of the random variables in each dimension, |i|=i1+...+i n The larger the value of l, the higher the accuracy; typically l is 2 or 3. After determining the sample, the coefficient matrix γ is solved by two least-squares weighted regressions:

[0063] γ T =(Ψ) T Ψ) -1 Ψ T ((Ξ T WΞ) -1 Ξ T WF) T (19)

[0064] in,

[0065]

[0066]

[0067] Step 4: Predict the dynamic response of the mixed uncertainty system. For mixed uncertainty systems that contain interval parameters and random parameters with non-normal distribution, improve the prediction accuracy of their dynamic response. At the same time, for mixed uncertainty systems that contain interval parameters and random parameters defined by an unknown distribution of data set, improve the prediction efficiency of their dynamic response.

[0068] For mixed uncertain systems containing interval parameters and random parameters with non-normal distributions, this invention can improve the accuracy of their dynamic response and reliability prediction; at the same time, for mixed uncertain systems containing interval parameters and random parameters defined by data sets with unknown distributions, this invention can improve the efficiency of their dynamic response and reliability prediction.

[0069] After obtaining the coefficient matrix γ, the interval mean and interval error bars of the response of the mixed uncertainty system are represented as follows:

[0070]

[0071]

[0072] in,

[0073]

[0074] Step 5: Predict the reliability of mixed uncertainty systems. For mixed uncertainty systems that contain interval parameters and random parameters with non-normal distributions, improve the prediction accuracy of their reliability. At the same time, for mixed uncertainty systems that contain interval parameters and random parameters defined by data sets with unknown distributions, improve the prediction efficiency of their reliability.

[0075] Through steps 1 to 3, a polynomial surrogate model is established for the mixed uncertain performance function, making the implicit limit state equation explicit. Therefore, the MCS method is directly used on the surrogate model equation of the performance function for reliability analysis.

[0076] The method also includes step 6: Based on steps 1 to 3, a weighted, data-driven, multinomial surrogate model with high approximation accuracy is established for the mixed uncertainty system. Since the method only depends on the statistical moment information of the random variables when establishing the multinomial surrogate model, and is independent of the distribution type of the random parameters, the analysis and prediction method is applied to engineering systems with interval parameters and random parameters with arbitrary distributions (including continuous, discrete, and unknown distributions) in the engineering field. Then, according to step 4, a high-precision prediction of the dynamic response of the system with mixed uncertainty parameters is achieved. Based on steps 1 to 3, a multinomial surrogate model of the mixed uncertainty performance function with high approximation accuracy can be established in just one step. Then, according to step 5, a high-efficiency and high-precision prediction of the reliability of the system with mixed uncertainty parameters is achieved.

[0077] Beneficial effects:

[0078] Regarding dynamic response prediction:

[0079] 1. When a mixed uncertainty system contains interval parameters and random parameters that follow a normal distribution, compared with traditional dynamic response prediction methods, the polynomial surrogate model of the mixed uncertainty system disclosed in this invention establishes the polynomial basis corresponding to the random variables in a data-driven manner, thus achieving higher prediction accuracy of the dynamic response.

[0080] 2. When a mixed uncertainty system contains interval parameters and random parameters with non-normal distribution, the polynomial basis corresponding to the random variable in the dynamic response prediction method disclosed in this invention is only related to the statistical moment information of the random parameters and is independent of the distribution type. Therefore, it is not necessary to convert the non-normally distributed random variable into a normal distribution, thereby avoiding the introduction of strong nonlinearity. As a result, its dynamic response prediction accuracy is higher.

[0081] 3. When a mixed uncertainty system contains interval parameters and random parameters with discrete distributions or defined by unknown distributions, the dynamic response prediction method disclosed in this invention uses a multinomial surrogate model to approximate the original mixed uncertainty system. The dynamic response of the mixed uncertainty system can be obtained by using the least squares method with double weighting. Moreover, the required sampling points are much fewer than those of traditional methods (nested scanning and MCS methods). Therefore, the dynamic response prediction method disclosed in this invention has higher prediction efficiency.

[0082] Regarding reliability prediction:

[0083] 1. The reliability prediction method for mixed uncertain systems disclosed in this invention can establish a polynomial surrogate model of the mixed uncertain performance function with high approximation accuracy in just one step, and can call the original performance function through parallel computing. In contrast, some current reliability analysis methods require the gradual updating of sampling points through learning functions or convergence conditions to make the surrogate model more accurate.

[0084] 2. The reliability prediction method for hybrid uncertainty systems disclosed in this invention calls the performance function only to establish a polynomial surrogate model, and then uses MCS for reliability analysis. Therefore, the time spent on the reliability analysis process is almost negligible.

[0085] 3. For mixed uncertain systems that simultaneously contain interval parameters and random parameters with unknown distributions, the reliability prediction method for mixed uncertain systems disclosed in this invention uses a weighted data-driven multinomial surrogate model to approximate the performance function of the original mixed uncertain system. The number of times the original performance function is called is much less than that of the MCS method. Therefore, the reliability prediction method disclosed in this invention has higher prediction efficiency. Attached image description:

[0086] Figure 1 This is a flowchart of the present invention;

[0087] Figure 2 To predict the upper bound (UB) and lower bound (LB) of the interval mean of the dynamic response X of the Duffing oscillator with mixed uncertainty parameters using three different calculation methods, including nested scan and Monte Carlo simulation (SMMSC), PCLM method and WDDPCLM method disclosed in this invention.

[0088] Figure 3 The upper bound (UB) and lower bound (LB) of the interval error bar for the dynamic response X of the Duffing oscillator with mixed uncertainty parameters predicted by the above three different calculation methods are given.

[0089] Figure 4 A histogram of the data distribution with random parameter α;

[0090] Figure 5 A histogram of the data distribution for the random parameter γ;

[0091] Figure 6 A histogram of the data distribution for random parameter h;

[0092] Figure 7 To predict the upper bound (UB) and lower bound (LB) of the interval mean of the dynamic response X of the Duffing oscillator with mixed uncertainty parameters by using two different calculation methods, including SMMSC and the WDDPCLM method disclosed in this invention.

[0093] Figure 8 The upper bound (UB) and lower bound (LB) of the interval error bar for the dynamic response X of the Duffing oscillator with mixed uncertainty parameters predicted by the above three different calculation methods are given.

[0094] Figure 9 This is a schematic diagram of the roof structure;

[0095] Figure 10 This is a schematic diagram of a space crank-slider;

[0096] Figure 11 This is a histogram of the data distribution for density ρ;

[0097] Figure 12 This is a histogram showing the data distribution of Young's modulus E.

[0098] Figure 13 This is a histogram of the data distribution for Poisson's ratio ν. Detailed Implementation

[0099] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.

[0100] Example 1:

[0101] This embodiment discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system. The specific implementation steps are as follows:

[0102] Predict the dynamic response of a Duffing oscillator with mixed uncertainties, where the oscillator contains interval parameters and random parameters following a normal distribution:

[0103] Step 1: Taking the Duffing oscillator as an example, the effectiveness of the method proposed in this invention will be illustrated. The governing equation of the Duffing oscillator is:

[0104]

[0105] The random parameters α, γ, and h follow a normal distribution:

[0106]

[0107] β and ω are interval parameters:

[0108] [β]~[2,4],[ω]~[98,102] (27)

[0109] Step 2: Establish the Legendre orthogonal polynomial Ψ of order q (q=3) corresponding to the interval variable. j([n])(j=0,...,T-1), where [n]~[-1,1].

[0110] Step 3: Based on the first 2p (p=3) order raw moments of the random variables, establish the p-order orthogonal polynomials corresponding to each dimension of the random variables.

[0111] Step 4: Establish a multidimensional orthogonal polynomial basis for the random variables.

[0112] The random variables in each dimension are normalized, resulting in a mean of 0 and a standard deviation of 1. Then, the orthogonal polynomial basis in each dimension constructed in step 3 is used to obtain a multidimensional orthogonal polynomial basis through tensor product operations.

[0113] Step 5: By going through steps 2 to 4, a surrogate model of the Duffing oscillator response with mixed uncertainties can be established.

[0114] Step 6: Determine the sample points. Samples for the interval uncertainty variable are selected using consistent coherent sampling techniques, with a sample size of 2J, where J = (q + m)! / (q!m!), and m = 2 is the dimension of the interval variable. Samples and weights of the random uncertainty variable are obtained through a sampling method similar to sparse grid numerical integration.

[0115] Step 7: Calculate the coefficient matrix in the polynomial surrogate model of the Duffing oscillator with mixed uncertainty based on two least squares regressions.

[0116] Step 8: Using nested scanning method and MCS method (SMMCS) as reference results, the interval mean and interval error bar of the dynamic response of the Duffing oscillator are calculated by formulas (22)-(24) of this invention, and the interval mean and interval error bar of the dynamic response of the Duffing oscillator are calculated by polynomial-chaos-Legendre-metamodel (PCLM) method.

[0117] Let x = X, Figure 1 In response to the interval mean of X, Figure 2 Based on the interval error bars for X, the numerical simulation results show that the dynamic response prediction method for mixed uncertain systems disclosed in this invention is almost in perfect agreement with the results of the nested scanning method and the MCS method (SMMCS), while the PCLM method has certain errors. This indicates that when the random parameters in the mixed uncertainty parameters follow a normal distribution, the method disclosed in this invention has higher prediction accuracy than traditional methods in predicting the dynamic response of mixed uncertain systems.

[0118] Example 2:

[0119] This embodiment discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system. The specific implementation steps are as follows:

[0120] Predict the numerical response of a simple mathematical example with mixed uncertainties, where some random parameters follow a non-normal distribution:

[0121] Step 1: A simple mathematical example model is as follows:

[0122]

[0123] Where x1, x2, and x3 are random parameters that follow uniform, normal, and exponential distributions, respectively:

[0124]

[0125] y1 and y2 are interval parameters, defined as follows:

[0126] [y1]~[1.5,3.5],[y2]~[8,12] (30)

[0127] Step 2: Establish the q-th order Legendre orthogonal polynomial Ψ corresponding to the interval variable (q=3). j ([n])(j=0,...,T-1), where [n]~[-1,1].

[0128] Step 3: Based on the first 2p-order raw moments of the random variables, establish the p-order orthogonal polynomials corresponding to each dimension of the random variables.

[0129] Step 4: Establish a multidimensional orthogonal polynomial basis for the random variables.

[0130] The random variables in each dimension are normalized, resulting in a mean of 0 and a standard deviation of 1. Then, the orthogonal polynomial basis in each dimension constructed in step 3 is used to obtain a multidimensional orthogonal polynomial basis through tensor product operations.

[0131] Step 5: Steps 2 to 4 establish the surrogate model of response m in the mathematical example with mixed uncertainty.

[0132] Step 6: Determine the sample points. Samples for the interval uncertainty variable are selected using consistent coherent sampling techniques, with a sample size of 2J, where J = (q + m)! / (q!m!), and m = 2 is the dimension of the interval variable. Samples and weights of the random uncertainty variable are obtained through a sampling method similar to sparse grid numerical integration.

[0133] Step 7: Calculate the coefficient matrix in the polynomial surrogate model of the mathematical example with mixed uncertainty based on two least squares regressions.

[0134] Step 8: Using nested scanning method and MCS method (SMMCS) as reference results, the interval mean and interval error bars of the numerical response m in the mathematical example are calculated using formulas (22)-(24) of this invention, and the interval mean and interval error bars of the numerical response m in the mathematical example are calculated using PCLM method. When calculating the interval mean and interval error bars of the numerical response m using PCLM method, the non-normally distributed random variable is transformed into a normally distributed random variable.

[0135] The calculation results of the three methods are given in Table 1. It can be concluded that, compared with the traditional PCLM method, when the order of the orthogonal polynomial corresponding to the random variable is p=1 and p=2, the upper bound (IM_UB), lower bound (IM_LB), upper bound (IEB_UB), and lower bound (IEB_LB) of the interval mean m obtained by the method disclosed in this embodiment are closer to the reference value (SMMCS method). This is because the process of transforming a non-normally distributed random variable into a normally distributed random variable further introduces errors, reducing the accuracy of the calculation results of the PCLM method. The response prediction method disclosed in this invention establishes a multinomial surrogate model based on a data-driven approach, thus resulting in higher accuracy of the calculation results. By further increasing the order of the orthogonal polynomial corresponding to the random variable, i.e. when p=3, the data in Table 1 shows that the upper bound (IM_UB), lower bound (IM_LB), upper bound (IEB_UB), and lower bound (IEB_LB) of the interval mean m obtained by the method disclosed in this embodiment are closer to the reference value (SMMCS method), which further illustrates that the method disclosed in this embodiment has high prediction accuracy.

[0136] Furthermore, the data in Table 1 also shows that the method disclosed in this embodiment has high computational efficiency. N in Table 1 call This represents the total number of times the original model is called. It can be concluded that when p=1, the method disclosed in this embodiment has higher computational efficiency than the traditional method.

[0137] Table 1 Comparison Results

[0138]

[0139] Example 3:

[0140] This embodiment discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system. The specific implementation steps are as follows:

[0141] Predict the dynamic response of a Duffing oscillator with mixed uncertainties, where the random parameters are defined by a dataset with an unknown distribution:

[0142] Step 1: Taking the Duffing oscillator as an example, the effectiveness of the method proposed in this invention will be illustrated. The control equation of the Duffing oscillator is shown in equation (25). Wherein, β and ω are interval parameters, as shown in equation (27). The random parameters α, γ, and h are all sets of data with unknown distributions, and their data distributions are as follows: Figure 3 , Figure 4 and Figure 5 As shown.

[0143] Step 2: Establish the Legendre orthogonal polynomial Ψ of order q (q=3) corresponding to the interval variable. j ([n])(j=0,...,T-1), where [n]~[-1,1].

[0144] Step 3: Based on the first 2p (p=3) order raw moments of the random variables, establish the p-order orthogonal polynomials corresponding to each dimension of the random variables.

[0145] Step 4: Establish a multidimensional orthogonal polynomial basis for the random variables.

[0146] The random variables in each dimension are normalized, resulting in a mean of 0 and a standard deviation of 1. Then, the orthogonal polynomial basis in each dimension constructed in step 3 is used to obtain a multidimensional orthogonal polynomial basis through tensor product operations.

[0147] Step 5: By going through steps 2 to 4, a surrogate model of the Duffing oscillator response with mixed uncertainties can be established.

[0148] Step 6: Determine the sample points. Samples for the interval uncertainty variable are selected using consistent coherent sampling techniques, with a sample size of 2J, where J = (q + m)! / (q!m!), and m = 2 is the dimension of the interval variable. Samples and weights of the random uncertainty variable are obtained through a sampling method similar to sparse grid numerical integration.

[0149] Step 7: Calculate the coefficient matrix in the polynomial surrogate model of the Duffing oscillator with mixed uncertainty based on two least squares regressions.

[0150] Step 8: Using the SMMCS method as a reference result, the interval mean and interval error bar of the dynamic response of the Duffing oscillator are calculated using formulas (22)-(24) of this invention.

[0151] Figure 6 The mean of the interval. Figure 7The interval error bars are used as an example. Numerical simulation results show that the method disclosed in this embodiment almost perfectly matches the results of the SMMCS method. Since the random parameters are from a dataset with an unknown distribution, the PCLM method fails in this case. This demonstrates that when a mixed uncertainty system contains interval parameters and random parameters defined by a dataset with an unknown distribution, the method disclosed in this embodiment has high prediction accuracy and efficiency.

[0152] Example 4:

[0153] This embodiment discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system. The specific implementation steps are as follows:

[0154] Predict the reliability of roof structures with mixed uncertainty parameters, where the mixed uncertainty performance function includes interval parameters and random parameters following a normal distribution:

[0155] Step 1: Taking the roof structure studied by Xiao et al. as an example, the effectiveness of the method proposed in this invention will be illustrated. The roof structure is as follows: Figure 8 As shown. The bottom boom and tension bars are made of steel, while the top boom and compression bars are reinforced with concrete. A uniformly distributed load Q is applied to the roof structure, which can be equivalently converted into a nodal load P = Ql / 4. At the top node C, the vertical deflection can be expressed as...

[0156]

[0157] Where A C Let A be the cross-sectional area of ​​the concrete. s E represents the cross-sectional area of ​​the reinforcing steel. C and E S Q, l, and E represent the Young's modulus of concrete and reinforcing steel, respectively. S and E C All are normally distributed random variables:

[0158] Q~N(20000,1600 2 ),l~N(12,0.24 2 (32)

[0159] E S ~N(1.2e11,(8.49e9) 2 ),E C ~N(3e10,(2.4e9) 2 (33)

[0160] A S and A C For interval variables:

[0161] [A S]~[0.00093,0.00095],[A C [0.033, 0.035] (34)

[0162] The performance function of the roof structure is:

[0163] G(X,Y)=0.025-Δ C (35)

[0164] Where 0.025 is the maximum allowable vertical deflection of the top node C.

[0165] Step 2: Establish the Legendre orthogonal polynomial Ψ of order q (q=3) corresponding to the interval variable. j ([n])(j=0,...,T-1), where [n]~[-1,1].

[0166] Step 3: Based on the first 2p-order raw moments of the random variables, establish the p-order orthogonal polynomials corresponding to each dimension of the random variables.

[0167] Step 4: Establish a multidimensional orthogonal polynomial basis for the random variables.

[0168] The random variables in each dimension are normalized, resulting in a mean of 0 and a standard deviation of 1. Then, the orthogonal polynomial basis in each dimension constructed in step 3 is used to obtain a multidimensional orthogonal polynomial basis through tensor product operations.

[0169] Step 5: Steps 2 to 4 will establish a proxy model for the vertical deflection of the roof structure with mixed uncertainties.

[0170] Step 6: Determine the sample points. Samples for the interval uncertainty variable are selected using consistent coherent sampling techniques, with a sample size of 2J, where J = (q + m)! / (q!m!), and m = 2 is the dimension of the interval variable. Samples and weights of the random uncertainty variable are obtained through a sampling method similar to sparse grid numerical integration.

[0171] Step 7: Calculate the coefficient matrix in the polynomial surrogate model of the mixed uncertainty roof structure performance function based on two least squares regressions.

[0172] Step 8: The maximum and minimum failure probabilities of the roof structure are obtained using the MCS method on the established polynomial surrogate model of the mixed uncertainty roof structure performance function. The results calculated directly using the MCS method are used as a reference, and the numerical results of several methods given by Xiao et al. are compared. The numerical results are shown in Table 2.

[0173] Table 2 Comparison Results

[0174]

[0175] The data calculated by the MCS, SSIS, FORM-UUA, ALK-HRA, AK-MCS, POAL-Kriging, and KA-SSIS methods in Table 2 are from the research of Xiao et al. From the data in Table 2, it can be seen that when all random parameters in the mixed uncertainty parameters follow a normal distribution, the maximum failure probability calculated by the DDPCLM method (p=2) disclosed in this embodiment is... The results obtained are exactly the same as those obtained by the MCS method, therefore its relative error is But minimum failure probability The results differ significantly from those obtained by the MCS method, with a relative error of [missing information]. Further increasing the expansion order, i.e., when p = 3, the maximum failure probability calculated by the method (DDPCLM) disclosed in this embodiment. Minimum failure probability The results obtained are very close to those obtained by the MCS method, with relative errors of respectively. and This demonstrates that the method disclosed in this embodiment has high computational accuracy.

[0176] Furthermore, the data in Table 2 also shows that the total number of times the method disclosed in this embodiment calls the original model, N. call The reliability prediction method relies on the number of parallel threads M on the server. Other reliability calculation methods, such as FORM-UUA, ALK-HRA, AK-MCS, POAL-Kriging, and KA-SSIS, require iterative updates of samples to make the surrogate model more accurate, and therefore cannot directly use parallel computing. The larger the number of parallel threads M on the server, the more obvious the advantage of the reliability prediction method disclosed in this embodiment in terms of computational efficiency. In summary, when the mixed uncertainty system contains interval parameters and random parameters following a normal distribution, the method disclosed in this embodiment has a high advantage in both computational accuracy and computational efficiency.

[0177] Example 5:

[0178] This embodiment discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system. The specific implementation steps are as follows:

[0179] Predict the reliability of mathematical examples with mixed uncertainties, where some random parameters do not follow a normal distribution:

[0180] Step 1: The performance function equation of the mathematical example model is:

[0181]

[0182] Where x1, x2, and x3 are random parameters that follow uniform, normal, and exponential distributions, respectively:

[0183]

[0184] y1 and y2 are interval parameters:

[0185] [y1]~[1.5,3.5],[y2]~[8,12] (38)

[0186] Step 2: Establish the Legendre orthogonal polynomial Ψ of order q (q=3) corresponding to the interval variable. j ([n])(j=0,...,T-1), where [n]~[-1,1].

[0187] Step 3: Based on the first 2p-order raw moments of the random variables, establish the p-order orthogonal polynomials corresponding to each dimension of the random variables.

[0188] Step 4: Establish a multidimensional orthogonal polynomial basis for the random variables.

[0189] The random variables in each dimension are normalized, resulting in a mean of 0 and a standard deviation of 1. Then, the orthogonal polynomial basis in each dimension constructed in step 3 is used to obtain a multidimensional orthogonal polynomial basis through tensor product operations.

[0190] Step 5: Steps 2 to 4 establish a polynomial surrogate model for the performance function of a mathematical example with mixed uncertainties.

[0191] Step 6: Determine the sample points. Samples for the interval uncertainty variable are selected using consistent coherent sampling techniques, with a sample size of 2J, where J = (q + m)! / (q!m!), and m = 2 is the dimension of the interval variable. Samples and weights of the random uncertainty variable are obtained through a sampling method similar to sparse grid numerical integration.

[0192] Step 7: Obtain the coefficient matrix in the polynomial surrogate model of the performance function of the mixed uncertainty mathematical example based on two least squares regression calculations.

[0193] Step 8: The maximum and minimum failure probabilities are obtained using the MCS method on the established polynomial surrogate model of the performance function of the mixed uncertainty mathematical example. The results calculated directly using the MCS method are used as a reference. At the same time, the random parameters that do not follow a normal distribution are transformed into a normal distribution. The results calculated using the MCS method on the polynomial surrogate model established by the traditional PCLM method are used for comparison. The numerical results are shown in Table 3.

[0194] Table 3 Comparison Results

[0195]

[0196] The data in Table 3 shows that, compared to the PCLM method, the maximum failure probability calculated by the DDPCLM (p=1) method disclosed in this embodiment is higher. The results obtained are very close to those obtained by the MCS method, with a relative error of [missing value]. Minimum failure probability The results differ significantly from those obtained using the MCS method. When p=2, the minimum failure probability... The results obtained are quite close to those obtained by the MCS method, with a relative error of [missing value]. Further increasing the expansion order, i.e., when p=3, the maximum failure probability calculated by the reliability analysis method (WDDPCLM) disclosed in this embodiment. Minimum failure probability The results obtained are very close to those obtained by the MCS method, with relative errors of respectively. and This demonstrates that the method disclosed in this embodiment has high computational accuracy.

[0197] In summary, when the random parameters in the mixed uncertainty parameters do not all follow a normal distribution, the method disclosed in this embodiment has high computational accuracy.

[0198] Example 6:

[0199] This embodiment discloses a method for predicting the dynamic response and reliability of a hybrid uncertainty system. The specific implementation steps are as follows:

[0200] Predict the reliability of a space crank-slider with mixed uncertainty parameters, where the random parameters are a dataset with an unknown distribution:

[0201] Step 1: Establish the dynamic model of the spatial crank-slider mechanism using the absolute nodal coordinate formula. A schematic diagram of the spatial crank-slider mechanism is shown below. Figure 9 As shown, both the crank and connecting rod are flexible beams, modeled using 4 and 8 ANCF fully parametric beam elements respectively. In this embodiment, the slider is massless. Table 4 lists the detailed characteristic parameters of the deterministic model.

[0202] Table 4 Characteristic parameters of spatial crank-slider mechanism

[0203]

[0204]

[0205] In this embodiment, density ρ, Young's modulus E, and Poisson's ratio v are random parameters, and these random parameters are a set of data with unknown distributions, the distributions of which are as follows: Figure 10 , Figure 11 and Figure 12As shown. The lengths of the crank and connecting rod are both interval parameters:

[0206] L1=1+5%×[η1], L2=2+5%×2[η2] (39)

[0207] Where [η1] ~ [-1,1], [η2] ~ [-1,1]. Whether the displacement of the slider's center of mass at a certain time t meets the specified requirements can be represented by the following performance function:

[0208] G(X,Y,t)=[S0(X,Y,t)]-S(X,Y,t) (40)

[0209] Where [S0(X,Y,t)] is the allowable displacement, and S(X,Y,t) is the actual displacement.

[0210] Step 2: Establish the Legendre orthogonal polynomial Ψ of order q (q=3) corresponding to the interval variable. j ([n])(j=0,...,T-1), where [n]~[-1,1].

[0211] Step 3: Based on the first 2p-order raw moments of the random variables, establish the p-order orthogonal polynomials corresponding to each dimension of the random variables.

[0212] Step 4: Establish a multidimensional orthogonal polynomial basis for the random variables.

[0213] The random variables in each dimension are normalized, resulting in a mean of 0 and a standard deviation of 1. Then, the orthogonal polynomial basis in each dimension constructed in step 3 is used to obtain a multidimensional orthogonal polynomial basis through tensor product operations.

[0214] Step 5: Steps 2 to 4 establish a polynomial surrogate model of the performance function of the crank-slider with mixed uncertainty space.

[0215] Step 6: Determine the sample points. Samples for the interval uncertainty variable are selected using consistent coherent sampling techniques, with a sample size of 2J, where J = (q + m)! / (q!m!), and m = 2 is the dimension of the interval variable. Samples and weights of the random uncertainty variable are obtained through a sampling method similar to sparse grid numerical integration.

[0216] Step 7: Calculate the coefficient matrix in the polynomial surrogate model of the performance function of the crank-slider in the mixed uncertainty space based on two least squares regression calculations.

[0217] Step 8: For the established polynomial surrogate model of the performance function of the crank-slider in a mixed uncertainty space, the MCS method is used to obtain the maximum and minimum failure probabilities. At t = 0.7s, the displacement of the slider in the X direction is 0.8844m. Here, it is assumed that the allowable displacements [S0(X,Y,t)] of the slider in the X direction at t = 0.7s are 1.0430m, 1.0990m, and 1.1090m, respectively. The minimum failure probability... All are 0, maximum failure probability The values ​​are 0.9977, 0.7483, and 0.6833, respectively. It can be seen that the value of the allowable displacement has a significant impact on the failure probability. The larger the allowable value, the smaller the failure probability and the greater the reliability.

[0218] According to Examples 1 to 6, high-precision prediction of the dynamic response and reliability of a hybrid uncertain system is achieved, and the analysis and prediction results can be applied to large and complex systems in the engineering field, thereby solving related uncertain response prediction and related engineering problems in the field of uncertain system dynamics and reliability prediction.

[0219] The above detailed description further illustrates the purpose, method, and beneficial effects of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A method for predicting the dynamic response and reliability of a hybrid uncertain system, characterized in that: Includes the following steps, Step 1: Represent the polynomial surrogate model of the mixed uncertainty system based on the PCLM method; Step 2: Establish the orthogonal polynomial basis for the random variables based on data-driven methods, and use Legendre polynomials as the orthogonal polynomial basis for the interval parameters. The one-dimensional orthogonal polynomial basis is represented by the raw moment information of the random variables; the tensor product operation is performed on the one-dimensional orthogonal polynomial basis of each dimension of the random variables to obtain the multidimensional orthogonal polynomial basis corresponding to the random variables. Step 3: Solving the coefficient matrix in the polynomial surrogate model of the mixed-uncertainty system by twice-weighted least squares ; Step 4: Predict the dynamic response of a mixed uncertainty system. For mixed uncertainty systems that contain interval parameters and random parameters with non-normal distributions, improve the prediction accuracy of their dynamic response; at the same time, for mixed uncertainty systems that contain interval parameters and random parameters defined by data sets with unknown distributions, improve the prediction efficiency of their dynamic response. Step 5: Predict the reliability of mixed uncertainty systems. For mixed uncertainty systems that contain interval parameters and random parameters with non-normal distributions, improve the prediction accuracy of their reliability; at the same time, for mixed uncertainty systems that contain interval parameters and random parameters defined by data sets with unknown distributions, improve the prediction efficiency of their reliability. The method also includes step 6: Based on steps 1 to 3, a weighted, data-driven, multinomial surrogate model with high approximation accuracy is established for the mixed uncertainty system. Since the method only depends on the statistical moment information of the random variables when establishing the multinomial surrogate model, and is independent of the distribution type of the random parameters, the prediction method is applied to engineering systems with interval parameters and arbitrarily distributed random parameters in the engineering field. Then, based on step 4, a high-precision prediction of the dynamic response of the system with mixed uncertainty parameters is achieved. Based on steps 1 to 3, a multinomial surrogate model of the mixed uncertainty performance function with high approximation accuracy can be established in just one step. Then, based on step 5, a high-efficiency and high-precision prediction of the reliability of the system with mixed uncertainty parameters is achieved.

2. The method of claim 1, wherein: Step 1 is implemented as follows: output response comprising dimensional random variable and dimensional interval variable , a polynomial proxy model of the response is established; The polynomial proxy model established in step 1 is shown in formula (1): (1) in, Interval parameters The function is represented as: (2) in, It is a one-dimensional orthogonal polynomial basis. orthogonal polynomial basis The order of , satisfying , For a one-dimensional orthogonal polynomial basis Constructed through tensor product operations, not exceeding Rank orthogonal polynomial; in equation (2) For Legendre polynomials, Representative coefficient matrix The elements, and , , Let be the order of the expansion of the orthogonal polynomial corresponding to the interval variable; the following establishes a one-dimensional orthogonal polynomial basis for the random variable.

3. The method for predicting the dynamic response and reliability of a hybrid uncertainty system as described in claim 2, characterized in that: Step 2 is implemented as follows: Step 2.1: First consider the first 3D random variables The corresponding one-dimensional orthogonal polynomial basis The definition is as follows: (3) in, To find the coefficients of the one-dimensional orthogonal polynomial, Let be the order of the one-dimensional orthogonal polynomial; for simplicity, the dimension identifier in formula (3) is omitted below. Then formula (3) can be simplified as follows: (4) when If it is a continuous random variable, then its i-th The first-order origin moment is: , (5) Since obtaining orthogonal bases depends on the moments of the input samples, we can consider continuous and discrete probability density functions, or even datasets with unknown distributions; if If it is a discrete random variable, then its i-th The first-order origin moment is: , If random variable It is only represented as a set of sampling points Then its first The first-order origin moment is approximately expressed as: , Statistical moments are quantitative measures describing the shape of a set of random samples or probability distributions; based on the information from the raw moments, we obtain: (6) For the coefficients in formula (6) It is obtained by inverting the matrix on the left side of equation (6), however, when the order... As the size increases, this matrix may become very ill-conditioned; therefore, consider alternative methods for performing matrix operations on the Hankel moment matrix; the Hankel moment matrix is ​​defined as: (7) If a random variable is defined by a sampled dataset, then the set of samples must be determined in the Hamburger sense, which means that the Hankel moment matrix is ​​a positive definite matrix, i.e. The matrix is ​​obtained by performing Cholesky decomposition on the above Hankel moment matrix: (8) Make For the matrix Find the reverse (9) According to Mysovskih theory, the matrix The elements in the formula (4) represent the orthogonal basis, that is: (10) To avoid inverting the matrix, a one-dimensional orthogonal basis is obtained using a three-term recursive formula: (11) The coefficients of the three recursive terms and satisfy: ， (12) in , Therefore, it can be observed that the coefficients in the three recursive formulas are all matrices. Elements in the matrix, thus avoiding the need for manipulation of the matrix. Find the inverse; Step 2.2: For a multidimensional input system, perform tensor product operations on the one-dimensional orthogonal polynomials corresponding to each dimension, i.e., construct a multidimensional orthogonal polynomial: (13) (14) in It is a multidimensional index that contains information on all possible combinations of products for each univariate basis function; furthermore, because These are the original random variables, and the order of magnitude of the samples for each dimension of the random variable may differ significantly, leading to an "ill-conditioned" matrix that affects computational accuracy and efficiency. Therefore, it is necessary to normalize the random variables for each dimension, that is, to normalize each dimension... ,have: (15) at this time The mean is 0 and the standard deviation is 1; then, the orthogonal polynomial basis for each dimension is constructed using steps 2.1-2. Finally, the multidimensional orthogonal polynomial basis is obtained through tensor product operation.

4. The method for predicting the dynamic response and reliability of a hybrid uncertainty system as described in claim 3, characterized in that: Step 3 is implemented as follows: The coefficient matrix is ​​solved using two least squares regressions. Since the selection of the sampling technique is crucial to the robustness of the least squares regression method, consistent coherent sampling is used for samples with interval parameters. For samples with random parameters, the optimal sampling points and weights can be calculated based on three recursive relationships. The specific implementation method is as follows: First, construct a symmetric tridiagonal Jacobi matrix: (16) The elements of this matrix are obtained through a three-term recurrence relation, and the matrix... It is positive definite; matrix The eigenvalues ​​are The roots of the polynomial of order n, and the weight of each root is: (17) in, Indicates the first The first component of the standard eigenvector corresponding to each eigenvalue; then, a sampling method similar to sparse grid numerical integration is used to obtain samples of random parameters and the weights corresponding to each group of samples; the 1D integration points are obtained from the eigenvalues ​​and eigenvectors of equation (16). and weight Then, a tensor product operation is performed on this 1-dimensional integral to obtain a product with... -Horizontal accuracy of 5D sparse grid points: (18) in, Multiple exponents corresponding to random variables in each dimension. ; The larger the value, the higher the precision, usually... Choose 2 or 3; after determining the sample, solve the coefficient matrix through two least squares weighted regressions. : (19) in, ， (20) , (21)。 5. The method for predicting the dynamic response and reliability of a hybrid uncertainty system as described in claim 4, characterized in that: Step 4 is implemented as follows: For mixed uncertain systems containing interval parameters and random parameters with non-normal distributions, this invention can improve the accuracy of their dynamic response and reliability prediction; at the same time, for mixed uncertain systems containing interval parameters and random parameters defined by data sets with unknown distributions, this invention can improve the efficiency of their dynamic response and reliability prediction. Obtain the coefficient matrix Then, the interval mean and interval error bars of the response of the mixed uncertainty system are represented as follows: (22) (23) in, (24)。 6. The method for predicting the dynamic response and reliability of a hybrid uncertainty system as described in claim 5, characterized in that: Step 5 is implemented as follows: Through steps 1 to 3, a multinomial surrogate model is established for the performance function of mixed uncertainty, making the implicit limit state equation explicit. Therefore, the MCS method is directly used on the surrogate model equation of the performance function to predict the reliability of the mixed uncertainty system. For mixed uncertainty systems containing interval parameters and random parameters with non-normal distribution, the prediction accuracy of its reliability is improved. At the same time, for mixed uncertainty systems containing interval parameters and random parameters defined by data sets with unknown distribution, the prediction efficiency of its reliability is improved.

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