Method, system and readable storage medium for estimating failure probability of mechanical structural system

Abstract

The present application provides a mechanical structure system failure probability estimation method, system and readable storage medium, after obtaining the failure data of the mechanical structure system, a Kriging surrogate model is constructed according to the failure data, the performance function value corresponding to the sample is predicted based on the Kriging surrogate model, and it is judged whether the model has further improvement space, in the learning and improvement process, the probability density of the limit state surface, the prediction uncertainty and the candidate point are considered at the same time, the number of adding points is reduced by calling as few as possible expensive finite elements, and lower cost, higher efficiency and more accurate failure probability estimation is realized.

Description

Technical Field

[0001] This invention belongs to the field of reliability assessment, and in particular, it is a method for estimating the failure probability of a mechanical structure system. Background Technology

[0002] A mechanical structure system is a system designed to meet specific functional requirements under certain safety constraints. Due to uncertainties such as the physical characteristics of the mechanical structure system (e.g., tolerances in component manufacturing) and environmental loads (e.g., extreme weather conditions), the use of a mechanical structure system may exceed its rated range. In such cases, the mechanical structure system may lose its integrity and fail to perform its intended function. Reliability analysis of mechanical structure systems analyzes their performance; that is, given an uncertainty model of the mechanical structure system, its environment, and loads, it quantitatively assesses the probability that the mechanical structure system will fail to meet specific functional requirements. Failure probability is a common quantitative indicator for conducting reliability analysis of mechanical structure systems.

[0003] Failure Probability (P) f Failure probability refers to the probability that a mechanical structure system will lose its intended function under specified conditions and within a specified time. It is an inherent property of mechanical structure systems and an important indicator in reliability analysis, including research, development, and performance evaluation. The estimation of failure probability is often calculated using the mechanical structure system performance function G(x), which is commonly referred to as the finite element method in engineering. The calculation method for failure probability is as follows:

[0004] P f =P{G(x)≤0}=∫ G(x)≤0 f(x)dx (1)

[0005] In the formula P fLet f(x) represent the failure probability of the mechanical structure system; x represents the performance factors affecting the mechanical structure system. Performance factors are those that influence the performance of the structural system, including the system's dimensions, material properties, and environmental loads. Different performance factors and their corresponding performance function values ​​constitute the failure data of the mechanical structure system. f(x) is the joint probability density function of x. The region containing factors with positive performance function values ​​is the safe region, and the region containing factors with non-positive performance function values ​​is the failure region. A continuous surface whose performance function values ​​approach 0 is called the limit state surface (LSS). Therefore, reliability assessment requires identifying the system's limit state, which is the critical state that distinguishes between system safety and failure. When this critical state is exceeded, the system no longer meets certain performance indicators, i.e., the system fails. The key to estimating the failure probability lies in the estimated performance function values ​​near the limit state surface.

[0006] With the development of modern science and technology, the number of components constituting mechanical structural systems is increasing, the scale of these systems is becoming larger, and the costs associated with research and development and production are rising. This makes the failure problem of mechanical structural systems increasingly important. Accurately estimating the failure probability of a mechanical structural system helps in timely understanding its operational status and making correct decisions.

[0007] Currently, in the field of mechanical structure reliability, the estimation method for the failure probability of mechanical structure systems mainly involves introducing an index function I(x) and Monte Carlo simulation (MCS) sampled values ​​P. mcs To replace the true value P f ,Right now:

[0008]

[0009]

[0010] In the formula, N mcs Let I(x) be the number of samples in the Monte Carlo simulation, and I(x) be a bivariate index function based on the performance function value. The main idea is to generate a large number of global sampling points that cover as many factor combinations as possible, use finite element method to estimate the performance function values ​​of the sampling points, and output these estimates instead of the true values ​​to calculate the failure probability. This sampling process is repeated iteratively until the termination condition of the failure probability evaluation index is met. The coefficient of variation (Cov) of the failure probability is... Pf Commonly used as a convergence evaluation metric for failure probability estimates:

[0011]

[0012] In practical engineering, as the complexity of mechanical structure systems increases, the repeated use of the finite element method (FEM) suffers from increased costs and limited solution efficiency. Existing technologies, such as subset simulation, importance sampling, and linear sampling, offer improved sampling strategies that reduce the number of FEM calls and can be effective. These methods often incorporate surrogate models, establishing a "black box" response model of the "input-output" relationship between a known finite set of performance influencing factors and their corresponding performance function values. This model predicts the failure probability at any sampling point for unknown performance function values. Commonly used surrogate models include support vector machines, response surface methodology, artificial neural networks, and Kriging. The relative error of the failure probability... It is often used as an evaluation metric for solving failure probability models.

[0013]

[0014] Meanwhile, modeling based solely on the existing partial initial sample set cannot adequately account for the extreme state surface. That is, small differences may lead to errors in the sign of the predicted performance function value approaching 0, ultimately resulting in a large deviation in the estimated failure probability. Active Learning (AL) methods, which interact with the external environment to enhance the understanding of structural systems, models, etc., are often used to correct surrogate models. Under the active learning framework, learning points are expected based on certain addition criteria, i.e., combinations of performance influencing factors that significantly improve the failure probability estimation effect. These are added to the existing sample set and the Kriging surrogate model is updated. The learning points are iterated repeatedly until the Kriging model converges. This process is summarized as the Active Learning Kriging (ALK) method. Among them, the addition criteria are also called learning functions. For example, the error probability under the Kriging surrogate model is represented by the U learning function, and the EFF learning function and information entropy H learning function are expected to be improved. The expressions are shown in equations (6) to (8):

[0015]

[0016]

[0017]

[0018] In the formula, and Let Φ(·) and φ(·) represent the predicted value and estimated standard deviation of the performance function value by the Kriging surrogate model, respectively, where Φ(·) and φ(·) represent the cumulative distribution function (CDF) and probability density function (PDF) of the standard normal distribution, respectively, and the coefficient ε is given by φ(·).EFF The regulating factor of the EFF function, and Proportional, usually taken It is evident that different learning functions characterize the increase of learning points from different perspectives.

[0019] Furthermore, considering that the key to estimating the failure probability lies in the predicted value near the limit state surface, that is, the predicted value of x satisfying the equation G(x) = 0. Its zero point has attracted much attention. When adding learning points, more consideration can be given to learning points where the performance function is 0. Therefore, the new function value in the failure domain is "folded" into the safety domain for consideration, that is, the prediction of the performance function value G(x) is changed to the prediction of its absolute value |G(x)|. At the same time, combined with the normality of the prediction results of the Kriging surrogate model, relevant studies have introduced the Folded Normal Distribution into the construction of the learning function addition criterion, such as the REIF learning function based on the expected improvement feasibility function of the Folded Normal Distribution.

[0020]

[0021]

[0022] In the formula, the empirical weight w = 2.

[0023] Regarding the folded normal distribution, for a random variable X ~ N(μ,σ) 2 If Y = |X|, Y ≥ 0, then Y follows a folded normal distribution, and its probability density function is:

[0024]

[0025] In the formula, cosh(·) represents the hyperbolic cosine function. Its cumulative distribution function is:

[0026]

[0027] In the formula, erf(·) is the error function, and its mean is:

[0028]

[0029] In the formula, Φ(·) is the same as in formula (7), and is the cumulative distribution function of the standard normal distribution, with variance:

[0030]

[0031] In engineering, the requirements for estimating the failure probability of mechanical structural systems often include both accuracy and efficiency in solution. However, current point-addition criteria under the active learning Kriging framework often consider regions where predicted values ​​approach 0 (near the limit state surface) or regions with large prediction variance (i.e., regions with high uncertainty). This can cause the point-addition range to get stuck in local clusters and fail to converge, leading to the inefficiency of calling finite element methods (FEM). Furthermore, as the influencing factors in complex structural systems such as components and subsystems in real-world mechanical problems become more high-dimensional and the limit state surfaces more irregular, the problem of excessive and costly FEM calls also arises. Summary of the Invention

[0032] To address the aforementioned technical problems in existing technologies, this invention proposes a method for estimating the failure probability of mechanical structure systems.

[0033] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0034] On the one hand, the present invention provides a method for estimating the failure probability of a mechanical structure system, comprising the following steps:

[0035] (S1) Determine the n performance factors affecting the mechanical structure system, the distribution characteristics of each performance factor, and the performance function, where n is greater than or equal to 1;

[0036] (S2) Based on the distribution characteristics of performance factors, N0 first samples are generated by Latin hypercube sampling, and each first sample contains sample values ​​of n performance factors.

[0037] (S3) Calculate the true performance function value corresponding to each first sample based on the performance function, and construct the initial sample set S0 based on each first sample and the true performance function value corresponding to each first sample.

[0038] (S4) Generate N1 second samples based on the distribution characteristics of performance factors, and construct an initial global sample pool S1. Each second sample contains sample values ​​of n performance factors.

[0039] (S5) Construct a Kriging proxy model between performance factors and performance function values ​​based on the current sample set S0;

[0040] (S6) Based on the current Kriging surrogate model, predict the performance function value corresponding to each second sample in the current global sample pool S1, and estimate the failure probability value of the mechanical structure system based on the predicted performance function value corresponding to each second sample in the current global sample pool S1.

[0041] (S7) Determine whether the current Kriging proxy model meets the set accuracy requirements. If it does, proceed to step (S8); if it does not, find the learning point in the global sample pool S1 based on the minimum value of the ELG2 learning function, calculate the true performance function value of the learning point based on the performance function, add the learning point and the true performance function value corresponding to the learning point to the sample set S0, and return to step (S5).

[0042] (S8) Determine whether the estimated failure probability value of the mechanical structure system meets the set failure probability evaluation index requirements. If it meets the requirements, proceed to step (S9). If it does not meet the requirements, generate N2 second samples based on the distribution characteristics of the performance factors, add them to the global sample pool S1, and return to step (S6).

[0043] (S9) Output the current Kriging surrogate model and the corresponding estimated failure probability value of the mechanical structure system. Based on the estimated failure probability value of the mechanical structure system, the reliability assessment of the mechanical structure system can be realized.

[0044] On the other hand, the present invention provides a failure probability estimation system for a mechanical structure system, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the following steps:

[0045] (S1) Determine the n performance factors affecting the mechanical structure system, the distribution characteristics of each performance factor, and the performance function, where n is greater than or equal to 1;

[0046] (S2) Based on the distribution characteristics of performance factors, N0 first samples are generated by Latin hypercube sampling, and each first sample contains sample values ​​of n performance factors.

[0047] (S3) Calculate the true performance function value corresponding to each first sample based on the performance function, and construct the initial sample set S0 based on each first sample and the true performance function value corresponding to each first sample.

[0048] (S4) Generate N1 second samples based on the distribution characteristics of performance factors, and construct an initial global sample pool S1. Each second sample contains sample values ​​of n performance factors.

[0049] (S5) Construct a Kriging proxy model between performance factors and performance function values ​​based on the current sample set S0;

[0050] (S6) Based on the current Kriging surrogate model, predict the performance function value corresponding to each second sample in the current global sample pool S1, and estimate the failure probability value of the mechanical structure system based on the predicted performance function value corresponding to each second sample in the current global sample pool S1.

[0051] (S7) Determine whether the current Kriging proxy model meets the set accuracy requirements. If it does, proceed to step (S8); if it does not, find the learning point in the global sample pool S1 based on the minimum value of the ELG2 learning function, calculate the true performance function value of the learning point based on the performance function, add the learning point and the true performance function value corresponding to the learning point to the sample set S0, and return to step (S5).

[0052] (S8) Determine whether the estimated failure probability value of the mechanical structure system meets the set failure probability evaluation index requirements. If it meets the requirements, proceed to step (S9). If it does not meet the requirements, generate N2 second samples based on the distribution characteristics of the performance factors, add them to the global sample pool S1, and return to step (S6).

[0053] (S9) Output the current Kriging surrogate model and the corresponding estimated failure probability value of the mechanical structure system. Based on the estimated failure probability value of the mechanical structure system, the reliability assessment of the mechanical structure system can be realized.

[0054] On the other hand, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, performs the following steps:

[0055] (S1) Determine the n performance factors affecting the mechanical structure system, the distribution characteristics of each performance factor, and the performance function, where n is greater than or equal to 1;

[0056] (S2) Based on the distribution characteristics of performance factors, N0 first samples are generated by Latin hypercube sampling, and each first sample contains sample values ​​of n performance factors.

[0057] (S3) Calculate the true performance function value corresponding to each first sample based on the performance function, and construct the initial sample set S0 based on each first sample and the true performance function value corresponding to each first sample.

[0058] (S4) Generate N1 second samples based on the distribution characteristics of performance factors, and construct an initial global sample pool S1. Each second sample contains sample values ​​of n performance factors.

[0059] (S5) Construct a Kriging proxy model between performance factors and performance function values ​​based on the current sample set S0;

[0060] (S6) Based on the current Kriging surrogate model, predict the performance function value corresponding to each second sample in the current global sample pool S1, and estimate the failure probability value of the mechanical structure system based on the predicted performance function value corresponding to each second sample in the current global sample pool S1.

[0061] (S7) Determine whether the current Kriging proxy model meets the set accuracy requirements. If it does, proceed to step (S8); if it does not, find the learning point in the global sample pool S1 based on the minimum value of the ELG2 learning function, calculate the true performance function value of the learning point based on the performance function, add the learning point and the true performance function value corresponding to the learning point to the sample set S0, and return to step (S5).

[0062] (S8) Determine whether the estimated failure probability value of the mechanical structure system meets the set failure probability evaluation index requirements. If it meets the requirements, proceed to step (S9). If it does not meet the requirements, generate N2 second samples based on the distribution characteristics of the performance factors, add them to the global sample pool S1, and return to step (S6).

[0063] (S9) Output the current Kriging surrogate model and the corresponding estimated failure probability value of the mechanical structure system. Based on the estimated failure probability value of the mechanical structure system, the reliability assessment of the mechanical structure system can be realized.

[0064] Compared with the prior art, the technical effects of the present invention are at least reflected in the following aspects:

[0065] This invention acquires failure data of mechanical structure systems and constructs a Kriging surrogate model based on the failure data, thereby achieving more efficient and accurate failure probability estimation.

[0066] Furthermore, by simultaneously considering the proximity to the limit state surface, prediction uncertainty, and probability density of candidate points, and by minimizing the use of expensive finite element methods and reducing the number of point additions, we can achieve lower cost, higher efficiency, and more accurate failure probability estimation. Attached Figure Description

[0067] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0068] Figure 1 This is a flowchart of an embodiment of the present invention;

[0069] Figure 2 This is a schematic diagram of a nonlinear undamped single-degree-of-freedom mass-spring system in one embodiment.

[0070] Figure 3 This is a line graph showing the function values ​​of the ELG2 function with added points in the example.

[0071] Figure 4This is a diagram illustrating the convergence process of solving the failure probability in the example.

[0072] Figure 5 This is a comparison chart of the predicted and actual values ​​of the performance function after sorting in the example. Detailed Implementation

[0073] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0074] One embodiment provides a method for estimating the failure probability of a mechanical structure system, comprising the following steps:

[0075] (S1) Determine the n performance factors affecting the mechanical structure system, the distribution characteristics of each performance factor, and the performance function, where n is greater than or equal to 1;

[0076] (S2) Based on the distribution characteristics of performance factors, N0 first samples are generated by Latin hypercube sampling, and each first sample contains sample values ​​of n performance factors.

[0077] (S3) Calculate the true performance function value corresponding to each first sample based on the performance function, and construct the initial sample set S0 based on each first sample and the true performance function value corresponding to each first sample.

[0078] (S4) Generate N1 second samples based on the distribution characteristics of performance factors, and construct an initial global sample pool S1. Each second sample contains sample values ​​of n performance factors.

[0079] (S5) Construct a Kriging proxy model between performance factors and performance function values ​​based on the current sample set S0;

[0080] (S6) Based on the current Kriging surrogate model, predict the performance function value corresponding to each second sample in the current global sample pool S1, and estimate the failure probability value of the mechanical structure system based on the predicted performance function value corresponding to each second sample in the current global sample pool S1.

[0081] (S7) Determine whether the current Kriging proxy model meets the set accuracy requirements. If it does, proceed to step (S8); if it does not, find the learning point in the global sample pool S1 based on the minimum value of the ELG2 learning function, calculate the true performance function value of the learning point based on the performance function, add the learning point and the true performance function value corresponding to the learning point to the sample set S0, and return to step (S5).

[0082] (S8) Determine whether the estimated failure probability value of the mechanical structure system meets the set failure probability evaluation index requirements. If it meets the requirements, proceed to step (S9). If it does not meet the requirements, generate N2 second samples based on the distribution characteristics of the performance factors, add them to the global sample pool S1, and return to step (S6).

[0083] (S9) Output the current Kriging surrogate model and the corresponding estimated failure probability value of the mechanical structure system. Based on the estimated failure probability value of the mechanical structure system, the reliability assessment of the mechanical structure system can be realized.

[0084] It is understood that this invention does not limit whether the initial sample set S0 or the initial global sample pool S1 is constructed first. Figure 1 The embodiment shown provides a method for estimating the failure probability of a mechanical structure system. First, N1 samples are generated based on the factor distribution characteristics to construct an initial global sample pool S1. Then, N0 samples are generated using Latin hypercube sampling, and their true performance function values ​​are calculated to construct an initial sample set S0. Based on the sample set S0, a Kriging surrogate model between performance factors and performance function values ​​is constructed. The subsequent steps are the same as those in the previous embodiment and will not be repeated here.

[0085] In step (S2), N0 first samples are generated using Latin hypercube sampling, where N0 is determined by the following formula:

[0086]

[0087] Where n is the number of performance factors.

[0088] When using Latin hypercube sampling, the range of values ​​for each performance factor must be defined in advance, and the distribution characteristics of each performance factor must be known. For example, for a performance factor x that follows a normal distribution, the range is N(μ,σ). 2 When the range of values ​​is not explicitly given, empirical values ​​[μ-5σ, μ+5σ] can be used.

[0089] In step (S4), N1 second samples are generated based on the distribution characteristics of the performance factor, and an initial global sample pool S1 is constructed. For example, for a performance factor x that follows a normal distribution, the pool size is approximately N(μ,σ). 2 N1 normally distributed random numbers are generated based on the values ​​of μ and σ.

[0090] Thus, both S0 and S1 contain combinations of performance factor values ​​generated by different sampling methods. The difference is that S0 also includes the actual performance function value obtained based on finite element calculation, while S1 does not contain the performance function value.

[0091] In one embodiment of the present invention, the kriging proxy model between the performance factor and the performance function value is as follows:

[0092]

[0093] In the formula, Let F be the predicted value of the performance function, and F be the sample trend function matrix, where β = [β1, β2, ..., β]. n ] T Let be the regression coefficients of the Kriging model, x be the values ​​corresponding to the n performance factors, and z(x) be the error term following a normal distribution with a mean of 0 and a variance of . Right now The covariance is cov(z(x) (i) ),z(x (j) )):

[0094]

[0095] f T (x) is the transpose of the trend function f(x) in the Kriging model, and it has different forms such as first-order, second-order, and higher-order. and They are the i-th first sample and the j-th first sample, respectively. and Let z(x) represent the values ​​of the n performance factors in the i-th first sample and the n performance factors in the j-th first sample, respectively. (i) ) and z(x (j) R represents the error terms corresponding to the i-th and j-th first samples, respectively; θ (x (i) ,x (j) ) is the correlation function between the i-th first sample and the j-th first sample, which includes the Gaussian correlation coefficient θ.

[0096]

[0097] θ k It is the Gaussian correlation coefficient related to the dimension n. and They are x (i) and x (j) The k-th factor in.

[0098] The Kriging surrogate model constructed in the above embodiments uses a Universal Kriging (UK) model, where the order of the trend function f(x) and the Gaussian correlation coefficient θ are pre-selected in the practical problem. The Kriging model in the above embodiments is mainly implemented based on the Dace toolbox of MATLAB, which can be understood and used by researchers in the art. Of course, those skilled in the art can also construct Kriging surrogate models in other ways.

[0099] Furthermore, in step (S6), the performance function value corresponding to each second sample in the current global sample pool S1 is predicted based on the current Kriging surrogate model. and prediction variance as follows:

[0100]

[0101]

[0102] In the formula, G is the abbreviation for the true performance function value G(x) of the performance factor. Let f(x) be the predicted value of the performance function of the Kriging surrogate model, and f(x) be the joint probability density function of x. For the parameter β = [β1, β2, ..., β n ] T The least squares estimate, i.e.

[0103]

[0104] for The least squares estimate, i.e.

[0105]

[0106] As is common knowledge in this field, r(x) and u(x) are both intermediate variables in the solution process of the Kriging model, and those skilled in the art can understand and use them.

[0107] R is the Gaussian correlation matrix, i.e.

[0108]

[0109] Based on the predicted performance function values ​​corresponding to each second sample in the current global sample pool S1, the failure probability of the mechanical structure system is estimated using Monte Carlo sampling. as follows:

[0110]

[0111] In the formula, N mcs N represents the number of sampling points in the Monte Carlo sampling. mcs ≤N1; This is the predicted performance function value for the second sample in the global sample pool.

[0112] In one embodiment, a method is proposed to determine whether the current Kriging surrogate model meets the set accuracy requirements. Specifically, this method calculates the relative error between the true value of the failure probability of the mechanical structure system and the estimated failure probability value of the mechanical structure system. When the relative error of the failure probability If the current Kriging proxy model meets the accuracy requirements, then it is determined that the accuracy requirements are not met.

[0113] Without loss of generality, if the true value of the failure probability of a mechanical structure system is known, the relative error between the true value and the estimated value of the failure probability can be directly calculated. Furthermore, the method for calculating the relative error between the true value and the estimated value of the failure probability can also be directly estimated using the method defined by formula (5) described in the background art.

[0114] In one embodiment, a method for finding learning points in the global sample pool S1 based on the minimum value of the ELG2 learning function is proposed, including the following steps:

[0115] By introducing an improved function L, its minimum value approaches the limit state surface, thus minimizing the mean function. Maximize variance at the same time

[0116] The improved function L expression is as follows:

[0117]

[0118] for Calculate the expectation of the improved function L:

[0119]

[0120] In the formula, and The performance function predictions and standard deviations of the Kriging surrogate model are given. That is, the mean of the folded normal distribution corresponding to the predicted value of the performance function of the Kriging surrogate model;

[0121] Define E(L) as ELG(x). Each time a point is added, candidate points satisfying min(ELG(x)) should be selected, i.e., the numerator. The smaller the value, the closer the candidate point is to the limiting state surface; the larger the denominator, the greater the uncertainty of the candidate point, and it should be given priority as the ideal learning point. Furthermore, since the probability density of candidate points is different, they should be given priority in regions with high probability density.

[0122] Taking into account the proximity to the limit state surface, prediction uncertainty, and probability density of candidate points, this embodiment adds a probability density function factor to ELG(x). Defined as ELG2(x), that is

[0123]

[0124] Φ(·) represents the cumulative distribution function of the standard normal distribution;

[0125] Find the learning point x in the global sample pool S1. * , where x * =arg min(ELG2(x)), which minimizes the use of expensive finite element methods, reduces the number of point additions, and achieves lower cost, higher efficiency, and more accurate failure probability estimation.

[0126] In one embodiment, a method is proposed for determining whether the estimated failure probability value of a mechanical structure system meets the requirements of a set failure probability evaluation index, as follows:

[0127] Calculate the coefficient of variation of the estimated failure probability value of the mechanical structure system, and compare the coefficient of variation of the estimated failure probability value of the mechanical structure system with the set value. If the coefficient of variation of the estimated failure probability value of the mechanical structure system is less than the set value, it is considered to meet the set failure probability evaluation index requirements; otherwise, it is considered not to meet the set failure probability evaluation index requirements.

[0128] The coefficient of variation of the failure probability value of the mechanical structure system is calculated using formula (4) recorded in the background art. Without loss of generality, the calculated coefficient of variation of the failure probability value of the mechanical structure system is compared with the empirical value of 5% of the coefficient of variation. If the coefficient of variation is less than 5%, it is considered to meet the requirements of the set failure probability evaluation index; otherwise, it is considered not to meet the requirements of the set failure probability evaluation index, that is, the model needs further improvement.

[0129] The following is a specific application example of the present invention:

[0130] like Figure 2 The figure shows a schematic diagram of a nonlinear undamped single-degree-of-freedom mass-spring system. This system withstands rectangular load pulses of random duration and amplitude, and exhibits different performance characteristics under the influence of different factors.

[0131] The performance function expression for the nonlinear undamped single-degree-of-freedom mass-spring system is as follows:

[0132] G(x) = 3r - |z max |

[0133] Where c1 and c2 are spring constants, m represents the mass, and the frequency of the nonlinear undamped single-degree-of-freedom mass-spring system is... r is the displacement, z max The maximum displacement that a nonlinear, undamped, single-degree-of-freedom mass-spring system can withstand is:

[0134]

[0135] Where F1 is the net force acting on the body, and t1 represents different times.

[0136] All six performance factors follow a normal distribution and are independent of each other. The specific parameters of the distribution are shown in Table 1.

[0137] Table 1 Example Performance Factor Parameter Table

[0138]

[0139] The failure data of the nonlinear undamped single-degree-of-freedom mass-spring system in this example consists of different combinations of six factors and their corresponding performance function values.

[0140] The failure probability prediction of the nonlinear undamped single-degree-of-freedom mass-spring system in this example is performed using the method described in the above embodiments. First, the number of samples in the initial sample set S0 is determined to be N0 = 28. Then, Lading hypercube is used to sample each performance factor, where the value range of the i-th performance factor is [μ]. i -5σ i ,μ i +5σ i ], and calculate the true performance function value corresponding to each sample to obtain the initial sample set S0.

[0141] Then, based on the normality of each performance factor, N1 = 1.0 × 10⁻⁶ is generated. 5 N global points are also used as Monte Carlo sampling points when estimating the failure probability, i.e., N mcs =N1=1.0×10 5 Thus, the global candidate pool S1 is obtained.

[0142] The solution was implemented using MATLAB programming, ultimately adding 42 learning points. The parameter values ​​of the constructed Kriging surrogate model are: θ = 0.1000, β = [-0.4815, -0.1924, -0.2357, 0.4842, -0.2644, 0.0821, 0.3803]. Based on the method of this invention, the estimated value of the structural system failure probability was obtained. The "true" failure probability obtained from Monte Carlo simulation sampling The relative error between the two is

[0143] During the solution process, the function values ​​of the ELG2 function for the added points are shown in [link to solution]. Figure 3 The convergence process of the failure probability is shown in [link to relevant documentation]. Figure 4 The comparison between the predicted and actual performance function values ​​after sorting from smallest to largest is shown in the figure. Figure 5 It can be seen that the two overlap significantly, meaning that the performance function value prediction effect is better.

[0144] If we follow the classic U, EFF, and H learning functions from existing research, and apply the Kriging framework for active learning, under the same S0 and S1 conditions, the final number of learning points increased is 61, 48, and 67, respectively, with estimated failure probabilities of 2.8333%, 2.6433%, and 2.8393%, respectively. Specific results are compared in Table 2. (Table T...) c This indicates the number of times the finite element method is called, i.e., the number of learning points added. and These represent the coefficient of variation and relative error of the failure probability estimate, respectively; NRMSE represents the normalized root mean square error.

[0145]

[0146] Where y i This represents the actual response value. The NRMSE is used to predict the response value. The range of NRMSE is [0,1]. The smaller the value, the smaller the error between the predicted result and the actual result, and the more accurate the prediction result.

[0147] Table 2 Comparison of Instance Failure Probability Estimation Results

[0148]

[0149]

[0150] Note: In Table 2, “-” indicates that the failure probability estimate of the MCS method is used as a comparison item and the corresponding value at that point has not been calculated.

[0151] It can be seen that, compared with the estimation results under the classical learning function in existing research, the failure probability results calculated according to the present invention require fewer learning points while maintaining similar estimation accuracy, thus reducing the cost of calling the expensive finite element method. Therefore, the results calculated by the present invention are more efficient and more accurate.

[0152] On the other hand, the present invention provides a mechanical structure system failure probability estimation system, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps of the mechanical structure system failure probability estimation method provided in any of the above embodiments. This mechanical structure system failure probability estimation system is a computer device, which may be a server. The computer device includes a processor, a memory, a network interface, and a database connected via a system bus. The processor of the computer device provides computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system, a computer program, and a database. The internal memory provides an environment for the operation of the operating system and computer program in the non-volatile storage medium. The database of the computer device stores sample data. The network interface of the computer device is used for communication with external terminals via a network connection.

[0153] On the other hand, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the steps of the mechanical structure system failure probability estimation method provided in any of the above embodiments.

[0154] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0155] Matters not covered in this invention are common knowledge.

[0156] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0157] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

[0158] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for estimating the failure probability of a mechanical structure system, characterized in that, Includes the following steps: (S1) Determine the factors affecting the mechanical structure system n Each performance factor, its distribution characteristics, and its performance function are described below. n Greater than or equal to 1; (S2) Based on the distribution characteristics of the performance factor, Latin hypercube sampling is used to generate... Each first sample contains [number] samples. n Sample values ​​of each performance factor; (S3) Calculate the true performance function value corresponding to each first sample based on the performance function, and construct an initial sample set based on each first sample and the true performance function value corresponding to each first sample. ; (S4) Generation based on the distribution characteristics of performance factors A second sample is used to construct the initial global sample pool. Each second sample contains n Sample values ​​of each performance factor; (S5) Based on the current sample set Construct a Kriging proxy model between performance factors and performance function values; (S6) Predict the current global sample pool based on the current Kriging surrogate model. The performance function values ​​corresponding to the second sample in each sample are based on the predicted current global sample pool. The performance function values ​​corresponding to each second sample are used to estimate the failure probability of the mechanical structure system. Among them, the current global sample pool is predicted based on the current Kriging surrogate model. The performance function values ​​corresponding to each second sample in the middle and prediction variance ,as follows: In the formula, The true performance function value of the performance factor abbreviation, The predicted value of the performance function of the Kriging surrogate model. yes The joint probability density function, For parameters The least squares estimate, i.e. for The least squares estimate, i.e. and These are all intermediate variables used in solving the Kriging model. R The Gaussian correlation matrix is, i.e. Based on the predicted current global sample pool The performance function values ​​corresponding to the second samples are used to estimate the failure probability of the mechanical structure system using Monte Carlo sampling. ,as follows: In the formula, The number of sampling points for Monte Carlo sampling. ; This is the predicted performance function value corresponding to the second sample in the global sample pool; (S7) Determine whether the current Kriging surrogate model meets the set accuracy requirements. If it does, proceed to step (S8); if it does not, then use the ELG2 learning function to find the minimum value in the global sample pool. Find learning points in the dataset, calculate the true performance function value of the learning points based on the performance function, and add the learning points and their corresponding true performance function values ​​to the sample set. Return to step (S5); (S8) Determine whether the estimated failure probability value of the mechanical structure system meets the set failure probability evaluation index requirements. If it does, proceed to step (S9). If it does not, regenerate based on the distribution characteristics of the performance factor. The second sample is added to the global sample pool. Return to step (S6); (S9) Output the current Kriging agent model and the corresponding estimated failure probability value of the mechanical structure system. Based on the estimated failure probability value of the mechanical structure system, the reliability assessment of the mechanical structure system can be realized.

2. The method for estimating the failure probability of a mechanical structure system according to claim 1, characterized in that, In step (S2), Determine using the following formula: 。 3. The method for estimating the failure probability of a mechanical structure system according to claim 1, characterized in that, In step (S5), the Kriging proxy model is as follows: In the formula, The predicted value of the performance function. For the sample trend function matrix, represents the regression coefficients of the Kriging model. for n The values ​​corresponding to each performance factor The error term follows a normal distribution with a mean of 0 and a variance of . ,Right now Covariance is : Trend function of Kriging model The transpose of has first-order, second-order, and higher-order forms. and The first i The first sample and the first j The first sample, and The first i In the first sample n The value corresponding to the performance factor and the first j In the first sample n The values ​​corresponding to each performance factor and The first i The first sample and the first j The error term corresponding to the first sample; It is the first i The first sample and the first j The first sample contains a Gaussian correlation coefficient. Related functions, Is with dimension n The relevant Gaussian correlation coefficient and They are and The first in One factor.

4. The method for estimating the failure probability of a mechanical structure system according to claim 1, 2, or 3, characterized in that, In step (S7), the method for determining whether the current Kriging surrogate model meets the set accuracy requirements is: calculating the relative error between the true value of the failure probability of the mechanical structure system and the estimated failure probability value of the mechanical structure system. When the relative error of the failure probability If the accuracy requirement is met, then the current Kriging proxy model is determined to meet the accuracy requirement; otherwise, the accuracy requirement is not met.

5. The method for estimating the failure probability of a mechanical structure system according to claim 4, characterized in that, In step (S7), the minimum value of the ELG2 learning function is found in the global sample pool. Finding learning points involves the following steps: Introduce the improved function L as follows: for Find the expectation of the improved function L: In the formula, and The performance function predictions and standard deviations of the Kriging surrogate model are given. That is, the mean of the folded normal distribution corresponding to the predicted value of the performance function of the Kriging surrogate model; Will Defined as and in Add probability density function factor to Defined as ,Right now The cumulative distribution function represents the standard normal distribution; In the global sample pool Find learning points ,in .

6. The method for estimating the failure probability of a mechanical structure system according to claim 4, characterized in that, In step (S8), it is determined whether the estimated failure probability value of the mechanical structure system meets the set failure probability evaluation index requirements. The method is as follows: Calculate the coefficient of variation of the estimated failure probability value of the mechanical structure system, and compare the coefficient of variation of the estimated failure probability value of the mechanical structure system with the set value. If the coefficient of variation of the estimated failure probability value of the mechanical structure system is less than the set value, it is considered to meet the set failure probability evaluation index requirements; otherwise, it is considered not to meet the set failure probability evaluation index requirements.

7. A failure probability estimation system for a mechanical structure system, comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method for estimating the failure probability of a mechanical structure system as described in claim 1.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the method for estimating the failure probability of a mechanical structure system as described in claim 1.

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