Structural interval reliability analysis method based on Chebyshev expansion truncation model

Abstract

The present invention discloses a structural interval reliability analysis method based on a Chebyshev expansion truncation model. The method first establishes a Chebyshev expansion truncation approximate model based on the general law of Chebyshev expansion coefficients, determines the Chebyshev test design points, solves the partial coefficients of the Chebyshev approximate model, and uses the approximate model to replace the original high-dimensional and complex functional function. Then, combined with a multi-factor full-level test design method and a discrete optimization algorithm, the upper and lower limits of the Chebyshev expansion truncation approximate model and the reliability of the structure are calculated. The present invention has good versatility and adaptability when using a surrogate function method to evaluate the reliability and safety of complex engineering structures in fields such as civil engineering, mechanical engineering, and aerospace based on interval analysis, greatly reducing the amount of calculation required for structural analysis and simulation during reliability analysis and improving computational efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of structural reliability analysis, in particular to the aspect of structural reliability analysis using a proxy model to approximate a real complex high-dimensional structural performance function, and specifically to a structural interval reliability analysis method based on a Chebyshev expansion truncation model. Background Art

[0002] Product structural reliability analysis is a crucial component of engineering product design in fields such as civil engineering, mechanical engineering, and aerospace. This analysis, driven by uncertainties in loads, material properties, and the manufacturing and use of product structures, is crucial for safety assessment and safe operation, as well as for improving key influencing factors and increasing safety reserves.

[0003] The functional functions that characterize the normal operating capacity or critical safety of large, complex structures or products in fields such as civil engineering, mechanical engineering, and aerospace are often highly nonlinear and implicitly expressed. Classical reliability analysis methods such as the second-order moment method and Monte Carlo method are either inaccurate or very time-consuming to calculate. This is especially true when large-scale numerical methods such as finite element methods are required for extensive analysis. These methods struggle to meet the efficiency and accuracy requirements of reliability analysis in engineering practice. Furthermore, the uncertainty parameters of product structures are crucial for structural reliability analysis, and proper handling of these uncertain parameters is a prerequisite for accurate structural reliability analysis. Traditional uncertainty models rely on sufficient samples to establish accurate probability distribution models, which is costly and presents challenges in engineering practice. Non-probabilistic interval models require only minimal sample information to determine the range of uncertain parameters.

[0004] The functional functions of actual engineering structures are complex and often have strong nonlinear and high-dimensional characteristics. Constructing proxy functions to approximate the real functional functions for structural or product reliability analysis in fields such as civil engineering, mechanical engineering, and aerospace, especially when combined with Monte Carlo simulation methods, can avoid a large amount of structural response analysis while ensuring good reliability analysis accuracy, greatly improving the efficiency of reliability analysis and gaining more and more attention and application in engineering practice. Summary of the Invention

[0005] The purpose of the present invention is to address the above-mentioned defects in the prior art and provide a structural interval reliability analysis method based on a Chebyshev expansion truncation model. This analysis method has strong universality and can be applied to structural interval reliability analysis of various nonlinear and high-dimensional functional functions. First, based on the general law of the Chebyshev expansion coefficient, a Chebyshev expansion truncation approximate model is established, the Chebyshev test design points are determined, and the various partial coefficients of the Chebyshev approximate model are solved. The approximate model is used to replace the original high-dimensional and complex functional function. Then, combined with a multi-factor full-level test design method and a discrete optimization algorithm, the upper and lower limits of the Chebyshev expansion truncation approximate model and the reliability of the structure are calculated. This method greatly improves the computational efficiency of product structural reliability analysis and the applicability of complex high-dimensional problems.

[0006] The purpose of the present invention can be achieved by taking the following technical solutions:

[0007] A structural interval reliability analysis method based on a Chebyshev expansion truncation model, the reliability analysis method comprising the following steps:

[0008] S1. Specify the product structure in the field to be analyzed and the function g(x) that reflects the normal working ability or critical state of safe working of the product structure in the field to be analyzed, x=(x1,…,x k ,…,x d ) is an interval variable vector, component x k =[a k ,b k ], k=1,…,d, select the smaller Chebyshev polynomial expansion highest order n, set the sub-item retention threshold parameter ξ, and the value of the variable division interval number N when solving the extreme value, where a k 、b k are the components x in the interval variable vector k The lower and upper limits of , d is the number of components in the interval variable vector, and the fields to be analyzed include civil engineering, mechanical electronics, and aerospace;

[0009] S2. Determine the number of interpolation points m of the Chebyshev expansion trigonometric function form, and set m=n+1;

[0010] S3, filter to meet the condition 0≤i1+i2+...+i d ≤n Chebyshev expansion polynomials constitute an approximate model, where i1,i2,...,i d =0,1,2...,n are the components x1,x2,...,x in the interval variable vector respectively. d The power of

[0011] S4, take one of the m interpolation points in each interval and combine them, and get m interval parameters d d-dimensional interpolation point set in the form of trigonometric functions Convert the interpolation point set Ω1 into a d-dimensional interpolation point set in the standard form of the polynomial in is the jth value of the kth interval variable in the form of trigonometric function k interpolation points, for The corresponding interpolation points in the standard form of the polynomial;

[0012] S5, substituting the interpolation points of the interpolation point set Ω2 into the performance function to obtain the structural response;

[0013] S6. Calculate the coefficients of the Chebyshev polynomial approximation model based on the structural response of the interpolation points Retain the terms with coefficients greater than ξ to obtain the final Chebyshev expansion truncated approximate model

[0014] S7, any k-th interval parameter x k Divide into N equal parts, and use the multi-factor full-level design method to obtain (N+1) d A set of test points Ω: Using discrete optimization algorithm in (N+1) d Search among the test points to obtain the Chebyshev expansion truncated approximate model The approximate values of the maximum and minimum values are used as the upper limit of the structural response function g max and the lower limit g min , and calculate the reliability index β of the structure according to the following formula:

[0015]

[0016] Furthermore, in step S1, the item retention threshold parameter ξ is set to 10 -10 , satisfying ξ>10 -10 The truncated approximate model composed of all coefficients is sufficient to better approximate the true functional response interval.

[0017] Furthermore, in step S2, the statistics of the interpolation points m in the form of trigonometric functions are calculated as m=n+1 while ensuring the requirements of calculation efficiency and calculation accuracy, which can ensure the approximate result of the Chebyshev truncation model.

[0018] Furthermore, in step S4, the interpolation point θ k,j and x in the standard form of the polynomial k,j Calculate as follows:

[0019]

[0020] Standardization transformation of interpolation points can ensure that all variables are analyzed and designed in a unified variable space for reliability analysis and design, and ensure the approximation accuracy of the Chebyshev truncation approximation model.

[0021] Furthermore, in step S6, the coefficients of the approximate model are calculated according to the following expression:

[0022]

[0023] in represents the coefficient of the Chebyshev polynomial approximation model, g(·) represents the performance function, Represents the tensor product of multiple one-dimensional Chebyshev polynomial triangular expansions, where i1,i2,...,i d =0,1,2...,n are the variables x1,x2,...,x in each sub-item respectively d The power of Represent the interpolation points The coefficients of the approximate model directly reflect the approximation results of the model, and the coefficients satisfying 0≤i1+i2+...+i d ≤n, and ignore the terms with smaller approximate model coefficients. On the one hand, the computational cost is greatly reduced, and on the other hand, the accuracy of the structural response can be controlled within the accuracy required by the structural reliability analysis.

[0024] Furthermore, in step S7, the upper limit g of the structural response function is max and the lower limit g min Calculate according to the following expression:

[0025]

[0026]

[0027] where max(·) and min(·) represent the maximum and minimum values. Compared with traditional optimization algorithms, the discrete optimization algorithm based on the multi-factor full-level experimental design method can improve efficiency and reduce computational costs in obtaining the response range of the structural performance function.

[0028] The present invention has the following advantages and effects compared to the prior art:

[0029] (1) The traditional Chebyshev expansion model is used to replace the performance function, and the interval reliability analysis is performed in combination with the optimization algorithm. The accuracy is very high, but the calculation amount is large. The present invention is based on the general law of the Chebyshev expansion coefficient, establishes a Chebyshev expansion truncation approximate model, ignores the terms that affect the accuracy, and only requires a small number of Chebyshev test design points to solve the various sub-item coefficients of the Chebyshev approximate model. The approximate model is used to replace the original high-dimensional and complex performance function for reliability analysis, which greatly reduces the number of calculations of the performance function.

[0030] (2) Chebyshev expansion interval analysis is based on the interval analysis of each sub-item, and does not fully consider the interdependence between the sub-items. Although the calculation efficiency is high, the error is large and even wrong interval analysis results are obtained. The present invention combines the multi-factor full-level test design method and the discrete optimization algorithm. On the basis of the Chebyshev expansion truncation approximation model, the approximate upper and lower limits and reliability of the structural response are calculated. The calculation efficiency is greatly improved, which can meet the accuracy requirements of the reliability analysis of engineering structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0032] Figure 1 This is a flow chart of a structural interval reliability analysis method based on a Chebyshev expansion truncation model disclosed by the present invention;

[0033] Figure 2 It is a schematic diagram of the I-shape in Example 2. DETAILED DESCRIPTION

[0034] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0035] Example 1

[0036] Figure 1 This is a flow chart of a structural interval reliability analysis method based on a Chebyshev expansion truncation model disclosed in this embodiment. This embodiment 1 further illustrates the present invention using an application example involving two interval uncertain parameters.

[0037] A structural interval reliability analysis method based on a Chebyshev expansion truncation model includes the following steps:

[0038] S1. Specify the structure to be analyzed, x1, x2 are interval variables, and the corresponding function g(x) is:

[0039] f(x)=(1.5x1-2) 2 -(x2-3) 2 +x1x2+10sin(2πx1)+10sin(2πx2)

[0040] Table 1 lists the upper and lower limits of each interval parameter. k 、b k (k=1,2), select a smaller Chebyshev polynomial expansion with the highest order n=3, and set the item retention threshold parameter ξ=10 -10 , when solving the extreme value, the number of intervals of variable division N=6;

[0041] Table 1. Parameter characteristics of each interval

[0042] Uncertainty <![CDATA[Lower limit b of the interval k > <![CDATA[Upper limit a of the interval k > type <![CDATA[x1]]> 4 5 interval variables <![CDATA[x2]]> 4 5 interval variables

[0043] S2. Determine the number of interpolation points m of the Chebyshev expansion trigonometric function form, and set m=n+1;

[0044] S3, filter to meet the condition 0≤i1+i2+...+i d ≤n Chebyshev expansion polynomials constitute an approximate model, where i1,i2,...,i d =0,1,2...,n are the variables x1,x2,...,x in each sub-item respectively d The power of

[0045] S4. Take one of the m interpolation points of each interval parameter and combine them. The d interval parameters will be m d d-dimensional interpolation point set in the form of trigonometric functions Convert it into a d-dimensional interpolation point set in the standard form of the polynomial in is the jth value of the kth interval variable in the form of trigonometric function k interpolation points, for The corresponding interpolation points in the standard form of the polynomial;

[0046] S5, substituting the interpolation point of Ω2 into the performance function to obtain the structural response;

[0047] S6. Calculate the coefficients of the Chebyshev polynomial approximation model based on the structural response of the interpolation points Retain the terms with coefficients greater than ξ to obtain the final Chebyshev expansion truncated approximate model

[0048] S7, any k-th interval parameter x k Divide into N equal parts, and use the multi-factor full-level design method to obtain (N+1) d A set of test points Ω:

[0049]

[0050] Using discrete optimization algorithm in this (N+1) d Search among the test points to obtain the Chebyshev expansion truncated approximate model The approximate values of the maximum and minimum values are used as the upper limit of the structural response function g max and the lower limit g min , and calculate the reliability index β of the structure according to the following formula:

[0051]

[0052] The structural response intervals and their relative errors calculated by the reliability analysis method disclosed in Example 1 and other methods are shown in Table 2. The upper and lower limits obtained by the continuous optimization method are taken as the exact solution, and the exact solution is [21.9382, 59.9454]. Table 2 compares the structural response intervals obtained when the Chebyshev approximation model order n = 3 and 5. When n = 3, the response interval calculated by the method of the present invention is [19.2341, 62.5882], and the relative errors of the upper and lower limits are 4.41% and 12.33%, respectively, and the performance function is calculated 8 times; when n = 5, the response interval calculated by the method of the present invention is [21.9959, 59.8851], and the relative errors of the upper and lower limits are reduced to 0.10% and 0.26%, respectively, and the performance function is calculated 10 times. The relative errors of the upper and lower limits of the Chebyshev subinterval method exceed 19% and 6%, respectively. It can be seen that the error of the method of the present invention is very small, the number of performance function calculations required is small, and the accuracy requirements of the structural interval reliability analysis are met.

[0053] Table 2. Response intervals and relative errors calculated by various methods in Example 1

[0054]

[0055] Example 2

[0056] This embodiment 2 further illustrates the present invention using an I-beam as an application example including eight interval variables. A structural interval reliability analysis method based on a Chebyshev expansion truncation model includes the following steps:

[0057] S1. Specify the structure to be analyzed. The corresponding function g(x) is:

[0058] M=g(P,L,a,S,d,b f ,t w ,t f )=σ max -S

[0059] Where x=(P,L,a,S,d,b f ,t w ,t f ),

[0060]

[0061]

[0062] Table 3 lists the upper and lower limits of each interval parameter. k 、b k (k=1,2), select a smaller Chebyshev polynomial expansion with the highest order n=3, and set the item retention threshold parameter ξ=10 -10 , when solving the extreme value, the number of intervals of variable division N=6;

[0063] Table 3. Parameter characteristics of each interval

[0064] Uncertainty <![CDATA[Lower limit b of the interval k > <![CDATA[Upper limit a of the interval k > type P 5670 6470 interval variables L 108 132 interval variables a 60 84 interval variables S 160480 179520 interval variables d 2.217 2.383 interval variables <![CDATA[b f ]]> 2.2 2.4 interval variables t 0.077 0.243 interval variables <![CDATA[t f ]]> 0.177 0.343 interval variables

[0065] S2. Determine the number of interpolation points m of the Chebyshev expansion trigonometric function form, and set m=n+1;

[0066] S3, filter to meet the condition 0≤i1+i2+...+i d ≤n Chebyshev expansion polynomials constitute an approximate model, where i1,i2,...,i d =0,1,2...,n are the variables x1,x2,...,x in each sub-item respectively d The power of

[0067] S4. Take one of the m interpolation points of each interval parameter and combine them. The d interval parameters will be m d d-dimensional interpolation point set in the form of trigonometric functions Convert it into a d-dimensional interpolation point set in the standard form of the polynomial in is the jth value of the kth interval variable in the form of trigonometric function k interpolation points, for The corresponding interpolation points in the standard form of the polynomial;

[0068] S5, substituting the interpolation point of Ω2 into the performance function to obtain the structural response;

[0069] S6. Calculate the coefficients of the Chebyshev polynomial approximation model based on the structural response of the interpolation points Retain the terms with coefficients greater than ξ to obtain the final Chebyshev expansion truncated approximate model

[0070] S7, any k-th interval parameter x k Divide into N equal parts, and use the multi-factor full-level design method to obtain (N+1) d A set of test points Ω:

[0071]

[0072] Using discrete optimization algorithm in this (N+1) d Search among the test points to obtain the Chebyshev expansion truncated approximate model The approximate values of the maximum and minimum values are used as the upper limit of the structural response function g max and the lower limit g min , and calculate the reliability index β of the structure according to the following formula:

[0073]

[0074] The comparison of the structural response ranges and relative errors calculated by the reliability analysis method disclosed in Example 2 and other methods is shown in Table 4. The upper and lower limits obtained by the continuous optimization method are taken as the exact solution, which is [-110533.0191, 115948.3707]. Table 4 compares the structural response intervals obtained when the Chebyshev approximation model order n=3 and 5. When n=3, the response interval calculated by the method of the present invention is [-110916.6452, 115523.9539], the relative errors of the upper and lower limits are 0.37% and 0.35%, and the performance function is calculated 112 times. When n=5, the response interval calculated by the method of the present invention is [-110543.9780, 116686.0451], the relative errors of the upper and lower limits are 0.64% and 0.01%, and the performance function is calculated 635 times. The relative errors of the upper and lower limits of the Chebyshev subinterval method exceed 20% and 10%, respectively. It can be seen that the error of the method of the present invention is very small, the number of performance function calculations required is small, and the accuracy requirements of the structural interval reliability analysis are met.

[0075] Table 4. Response intervals and relative errors calculated by various methods in Example 2

[0076]

[0077] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A structural interval reliability analysis method based on Chebyshev expansion truncation model, characterized in that: The reliability analysis method comprises the following steps: S1. Specify the product structure in the field to be analyzed and the function g(x) that reflects the normal working ability or critical state of safe working of the product structure in the field to be analyzed, x=(x1,…,x k ,…,x d ) is an interval variable vector, component x k =[a k ,b k ], k=1,…,d, select the smaller Chebyshev polynomial expansion highest order n, set the sub-item retention threshold parameter ξ, and the value of the variable division interval number N when solving the extreme value, where a k 、b k are the components x in the interval variable vector k The lower and upper limits of , d is the number of components in the interval variable vector, and the fields to be analyzed include civil engineering, mechanical electronics, and aerospace; S2. Determine the number of interpolation points m of the Chebyshev expansion trigonometric function form, and set m=n+1; S3, filter to meet the condition 0≤i1+i2+...+i d ≤n Chebyshev expansion polynomials constitute an approximate model, where i1,i2,...,i d =0,1,2...,n are the components x1,x2,...,x in the interval variable vector respectively. d The power of S4, take one of the m interpolation points in each interval and combine them, and get m interval parameters d d-dimensional interpolation point set in the form of trigonometric functions Convert the interpolation point set Ω1 into a d-dimensional interpolation point set in the standard form of the polynomial in is the jth value of the kth interval variable in the form of trigonometric function k interpolation points, for The corresponding interpolation points in the standard form of the polynomial; S5, substituting the interpolation points of the interpolation point set Ω2 into the performance function to obtain the structural response; S6. Calculate the coefficients of the Chebyshev polynomial approximation model based on the structural response of the interpolation points Retain the terms with coefficients greater than ξ to obtain the final Chebyshev expansion truncated approximate model S7, any k-th interval parameter x k Divide into N equal parts, and use the multi-factor full-level design method to obtain (N+1) d A set of test points Ω: Using discrete optimization algorithm in (N+1) d Search among the test points to obtain the Chebyshev expansion truncated approximate model The approximate values of the maximum and minimum values are used as the upper limit of the structural response function g max and the lower limit g min , and calculate the reliability index β of the structure according to the following formula:

2. The structural interval reliability analysis method based on the Chebyshev expansion truncation model according to claim 1 is characterized in that: In step S4, the interpolation point And the polynomial standard form Calculate as follows:

3. The structural interval reliability analysis method based on the Chebyshev expansion truncation model according to claim 1 is characterized in that: In step S5, the coefficients of the approximate model are calculated according to the following expression: in represents the coefficient of the Chebyshev polynomial approximation model, g(·) represents the performance function, Represents the tensor product of multiple one-dimensional Chebyshev polynomial triangular expansions, where i1,i2,...,i d =0,1,2...,n are the variables x1,x2,...,x in each sub-item respectively d The power of Represent the interpolation points The weight.

4. The structural interval reliability analysis method based on the Chebyshev expansion truncation model according to claim 1 is characterized in that: In step S6, the upper limit g of the structural response function max and the lower limit g min Calculate according to the following expression: Among them, max() and min() represent the maximum and minimum values.

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