An efficient and high-precision evaluation method for crack propagation life of aviation structure considering parameter correlation

Abstract

The present application relates to the technical field of computer-aided design, and particularly relates to a kind of efficient high-precision evaluation method of aviation structure crack propagation life considering parameter correlation, comprising: establishing mixed mode fatigue crack propagation finite element simulation model to obtain the deterministic fatigue life function of aviation structure;Identify uncertain input parameter vector and its edge distribution and statistical correlation;When there is correlation, map the correlated parameters to independent standard uniform variables based on Rosenblatt transformation and convert the independent variable fatigue life function of corresponding uniform distribution variable;In the independent space, the life function is decomposed into single variable function product by multiplication dimension reduction method, and the fractional moment is calculated by Gaussian quadrature;With fractional moment as constraint, reconstruct the fatigue life probability density function based on maximum entropy principle.The present application significantly reduces the cost of finite element calling, accurately captures long-tail life distribution, and effectively handles the statistical correlation between input parameters.

Description

Technical Field

[0001] This invention relates to the field of computer-aided design technology, and in particular to an efficient and high-precision method for evaluating the crack propagation life of aerospace structures that considers parameter-related factors. Background Technology

[0002] During service, structural components in aerospace and high-end equipment often endure combined Type I (opening) and Type II (slipping) fatigue loads caused by complex loads. This leads to crack initiation and propagation, becoming one of the main failure modes affecting structural integrity and safe lifespan. In existing technologies, Monte Carlo simulation is considered the standard approach to obtaining the lifespan probability distribution. This method involves large-scale sampling of input random variables and repeated calculations using a high-fidelity finite element model, ultimately approximating the overall distribution through sample statistical results. Although Monte Carlo simulation has high statistical accuracy when the sample size is sufficient, the number of calls to the finite element model typically reaches tens or even hundreds of thousands, resulting in extremely long computation times, making it difficult to meet the efficiency requirements of iterative engineering design or real-time condition assessment. To reduce computational overhead, surrogate model techniques such as the Kriging model or polynomial chaotic expansion are used to establish approximate response relationships between input parameters and fatigue life, thus replacing expensive finite element analysis. However, such methods often struggle to guarantee accuracy in the tail region of the probability distribution, and the number of training samples required to build a high-precision surrogate model increases exponentially as the dimensionality of the input variables increases, which is known as the curse of dimensionality.

[0003] While the aforementioned methods have achieved certain results within their respective applicable ranges, existing hybrid-mode fatigue crack propagation life probability assessments still face critical technical challenges that urgently need to be addressed: Firstly, the statistical correlation between input random variables is often ignored. Experimental observations show that there is a generally significant statistical correlation between material parameters such as the Paris coefficient C and the Paris exponent M in the Paris formula. However, most existing analysis procedures are based on the assumption of mutual independence of variables for modeling and solving. Forcibly assuming statistical independence will lead to a systematic shift in the reconstructed fatigue life probability distribution, especially in the low failure probability tail region, which characterizes the high reliability requirements of the structure. This may produce orders-of-magnitude non-conservative estimation errors, directly jeopardizing the credibility of the safety assessment conclusions. Secondly, while pursuing computational efficiency, it is difficult to simultaneously achieve high-precision characterization of the heavy-tailed characteristics of the fatigue life distribution. Fatigue life typically exhibits a significant skewed and heavy-tailed distribution. Traditional distribution reconstruction methods based on the first four integer moments have limited ability to extract information from the tail of the distribution and cannot sensitively capture the probability of extreme events. Therefore, there is an urgent need to develop an efficient and high-precision evaluation method that can accurately handle arbitrary correlations between input parameters and reconstruct the full probability distribution characteristics of fatigue life with high fidelity, under the premise of strictly controlling the number of finite element model calls.

[0004] Chinese Patent Publication No. CN114218661A discloses a fatigue life prediction method based on fatigue crack propagation, including: preprocessing, iterative solution, and crack propagation, which can be completed using SESAM, ABAQUS, and FRANC 3D platforms, respectively. The specific steps include: S1 modeling, S2 hydrodynamic analysis, S3 structural strength analysis, S4 three-dimensional crack propagation zone modeling, S5 crack insertion and mesh update, S6 fatigue characteristic characterization, S7 incremental calculation judgment, S8 termination condition verification, and S9 service life statistics. It can be seen that the fatigue life prediction method based on fatigue crack propagation has the following problems: the method focuses on crack propagation path tracking and life integration under a deterministic framework. Its input parameters, including initial crack size, material constants, and load environment parameters, are all regarded as deterministic quantities. It does not involve the quantitative characterization of the impact of parameter uncertainty on the life prediction results, and therefore cannot give a fatigue life distribution range and reliability assessment conclusion with probabilistic significance. Secondly, this method does not provide a mechanism for handling the statistical correlation between input random variables. In actual material performance testing, the coefficient C and the exponent M of the Paris formula describing crack propagation rate often show a significant statistical correlation. If this correlation structure is ignored and probability analysis is performed directly, the tail characteristics of the reconstructed lifetime distribution will show a systematic deviation, affecting the accuracy of safety assessment under high reliability design requirements. Summary of the Invention

[0005] To address this, the present invention provides an efficient and high-precision method for evaluating the crack propagation life of aerospace structures that considers parameter correlation, thereby overcoming the problems of neglecting parameter correlation and insufficient accuracy in reconstructing the heavy-tail region of the life distribution in the prior art.

[0006] To achieve the above objectives, this invention provides an efficient and high-precision method for evaluating the crack propagation life of aerospace structures, considering parameter-related factors, comprising: Step S1: Define the uncertain input parameter vector that affects the fatigue life of the structural component to be evaluated, and determine the statistical correlation between the uncertain input parameter vectors. Step S2: Based on finite element analysis, establish a finite element simulation model of fatigue crack propagation of the structural component to be evaluated under the combined action of type I and type II loads. Determine the deterministic fatigue life function of the structural component to be evaluated based on the fatigue crack propagation finite element simulation model and the uncertain input parameters. Step S3: Determine the independent variable function with respect to the uniformly distributed variable vector based on the deterministic fatigue life function, where: Based on the judgment result that there is a statistical correlation between the uncertain input parameter vectors, the Rosenblatt transformation is used to map the statistically correlated uncertain input parameter vectors into statistically independent uniformly distributed variable vectors. Based on the judgment result of statistical independence between the uncertain input parameter vectors, the uniformly distributed variable vectors that are statistically independent of each other are determined according to the uncertain input parameter vectors. Step S4: Decompose the independent variable function using the multiplicative dimensionality reduction method, and calculate the fractional moments of the lifetime response based on Gaussian quadrature. Step S5: Construct an optimization model based on the constraints of the fractional moments as the maximum entropy principle, solve for the Lagrange multipliers and fractional order, and obtain the probability density function of the fatigue life of the structural component to be evaluated.

[0007] Further, step S2 includes: Step S21: Based on finite element analysis, establish a finite element simulation model of fatigue crack propagation of the structural component to be evaluated under the combined action of type I and type II loads; Step S22: Solve the J integral using the region integration method to obtain the energy release rate; Step S23: Based on the fatigue crack propagation finite element simulation model and the energy release rate, determine the Type I stress intensity factor and Type II stress intensity factor of the structural component to be evaluated through interaction integration; Step S24: Determine the crack propagation direction based on the Type I stress intensity factor and the Type II stress intensity factor; Step S25: Calculate the equivalent stress intensity factor range based on the Type I stress intensity factor and the Type II stress intensity factor; obtain the deterministic fatigue life function based on the equivalent stress intensity factor range.

[0008] Furthermore, in step S24, the circumferential stress is determined based on the maximum circumferential stress criterion, and the crack propagation direction is determined according to the circumferential stress and the Type I stress intensity factor and the Type II stress intensity factor.

[0009] Furthermore, in step S25, the total fatigue crack propagation life in the deterministic fatigue life function is determined by adding the cyclic increments of the crack propagation history.

[0010] Furthermore, the uncertain input parameter vector includes the initial crack size, load amplitude, Paris coefficient, and Paris exponent.

[0011] Further, step S4 includes: Step S41: Select a reference point in the space of independent variables; Step S42: Use multiplicative dimensionality reduction approximation to decompose the independent variable function into a product of multiple univariate functions; Step S43: Select a set of fractional orders and calculate the fractional moments of the lifetime response based on Gaussian quadrature.

[0012] Furthermore, in step S43, the order of each fraction is a non-integer.

[0013] Further, step S5 includes: Step S51: Using the normalization condition and the fractional moments as constraints, construct a maximum entropy optimization model with the goal of maximizing Shannon entropy. Step S52: Construct the Kullback-Leibler divergence between the true probability density function of fatigue life and the estimated probability density function of fatigue life; Step S53: Solve for the minimized Lagrange multipliers and fractional order to minimize the Kullback-Leibler divergence; Step S54: Determine the probability density function of the fatigue life of the structural component to be evaluated based on the Lagrange multipliers.

[0014] Further, in step S53, the optimization procedure for minimizing the Lagrange multipliers and the fractional order is as follows: ; Where I is the objective function; Y is the crack propagation life random variable; and y is the realization of the crack propagation life random variable Y. It is a fractional moment; For Lagrange multipliers; Let be the fractional order; m is the number of fractional moments, i = 1, 2, ..., m.

[0015] Furthermore, the support set of the crack propagation life random variable is Ω = (0, ∞).

[0016] Compared with existing technologies, the beneficial effects of this invention are as follows: by transforming statistically correlated uncertain input parameters into independent uniform variables, combining multiplicative dimensionality reduction and Gaussian quadrature to calculate fractional moments, and using the fractional moments as constraints to reconstruct the lifetime probability density function using the maximum entropy principle, this invention can accurately obtain the complete probability distribution of fatigue crack propagation lifetime in mixed modes with very few finite element analyses. This significantly reduces computational costs, effectively preserves the correlation between parameters, and accurately captures the long-tail characteristics of the lifetime distribution, solving the problems of low computational efficiency, insufficient tail accuracy, and neglect of parameter correlation in traditional methods.

[0017] Furthermore, this invention maps statistically correlated input parameters into mutually independent uniformly distributed variables through conditional cumulative distribution inverse transformation. While preserving the true correlation structure between parameters, it performs probability analysis, avoiding the lifetime distribution estimation bias caused by the traditional independence assumption, and improving the accuracy of probability assessment and engineering credibility.

[0018] Furthermore, this invention employs a multiplicative dimensionality reduction method to approximately decompose the multidimensional lifetime function into the product of multiple single-variable functions, and utilizes the Gaussian quadrature rule to perform one-dimensional integration in a uniform space, which significantly reduces the number of calls to the high-fidelity finite element model and significantly improves computational efficiency, enabling precise probabilistic fatigue life assessment to be completed quickly under conventional computing resources.

[0019] Furthermore, this invention uses fractional moments as constraints for the maximum entropy principle. Compared with traditional integer moment methods, fractional moments can more effectively capture the skewness and heavy-tailed characteristics of fatigue life distribution, avoiding the problem of tail information distortion caused by increased estimation errors of integer moments. Thus, high-precision probability density estimation can be obtained in the low failure probability region, meeting the requirements of high reliability design.

[0020] Furthermore, by establishing a finite element model of crack propagation under combined type I and type II loads, the present invention uses the maximum circumferential stress criterion to determine the crack propagation direction and accumulates the lifetime cycle by cycle based on the equivalent stress intensity factor range, which can realistically reflect the physical process of crack bending propagation under complex load paths, ensuring the high fidelity of deterministic lifetime calculation and laying a reliable foundation for probabilistic analysis. Attached Figure Description

[0021] Figure 1 A flowchart illustrating a parameter-dependent, high-precision method for evaluating the crack propagation life of aerospace structures, as described in this invention. Figure 2 A comparison of the fatigue life probability density function of the center crack plate in the embodiment of the present invention, considering the parameter-dependent high-efficiency and high-precision evaluation method for crack propagation life of aerospace structures. Figure 3 A comparison of the survival probability curves of the center crack plate in the efficient and high-precision evaluation method for the crack propagation life of aerospace structures considering parameter-related factors in this embodiment of the invention. Figure 4 This is a comparison chart of the survival probability of different preset lifetime thresholds when considering the correlation coefficient changes of the parameter-dependent high-efficiency and high-precision evaluation method for the crack propagation lifetime of aerospace structures in an embodiment of the present invention. Figure 5 A refined finite element mesh of an aluminum bracket with an angular crack, representing an embodiment of the present invention, demonstrating an efficient and high-precision method for evaluating the crack propagation life of aerospace structures based on parameter-related factors. Figure 6 A comparison diagram of the variation of the stress intensity factor of the support along the crack propagation path in the efficient and high-precision evaluation method for crack propagation life of aerospace structures considering parameter-related factors in this embodiment of the invention. Figure 7 Comparison of fatigue life probability density functions of support structures using a parameter-dependent, high-precision method for evaluating crack propagation life of aerospace structures in this embodiment of the invention. Figure 8 The fatigue life probability density function of the support under different correlation coefficients of the efficient and high-precision evaluation method for crack propagation life of aerospace structures with parameter correlation is considered in the embodiments of the present invention. Figure 9 The survival probability curves of the support under different correlation coefficients of the efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation are shown in the embodiments of the present invention. Detailed Implementation

[0022] To make the objectives and advantages of the present invention clearer, the present invention will be further described below with reference to embodiments; it should be understood that the specific embodiments described herein are merely for explaining the present invention and are not intended to limit the present invention.

[0023] Please see Figure 1 The flowchart shown is a flowchart of the efficient and high-precision evaluation method for crack propagation life of aerospace structures considering parameter dependence according to the present invention. The present invention provides an efficient and high-precision evaluation method for crack propagation life of aerospace structures considering parameter dependence, comprising: Step S1: Define the uncertain input parameter vector that affects the fatigue life of the structural component to be evaluated, and determine the statistical correlation between the uncertain input parameter vectors. Specifically, the uncertain input parameter vector includes the initial crack size, load amplitude, Paris coefficient, and Paris exponent.

[0024] In this embodiment, C and M are modeled as correlated log-normal random variables and described by standardized Gaussian variables. C is the Paris coefficient, whose value is determined by the material experimental data of the structural component to be evaluated; M is the Paris exponent, whose value is determined by the material experimental data of the structural component to be evaluated. Furthermore, let ρ be the correlation coefficient between C and M. The correlation coefficient ρ varies in the interval [-1, 1] and is used to quantify the influence of statistical correlation on the reconstructed fatigue life distribution. The endpoint case ρ=±1 is numerically interpreted as the limiting case of the non-degenerate model, that is, the case when |ρ| approaches 1.

[0025] Step S2: Based on finite element analysis, establish a finite element simulation model of fatigue crack propagation of the structural component to be evaluated under the combined action of type I and type II loads. Determine the deterministic fatigue life function of the structural component to be evaluated based on the fatigue crack propagation finite element simulation model and the uncertain input parameters. Specifically, step S2 includes: Step S21: Based on finite element analysis, establish a finite element simulation model of fatigue crack propagation of the structural component to be evaluated under the combined action of type I and type II loads; In this embodiment, ABAQUS finite element software is used to establish a fatigue crack propagation finite element simulation model of the structure under evaluation under combined Type I and Type II loads. Specifically, a two-dimensional plane stress finite element model is established based on the actual geometric dimensions of the structure under evaluation. Adaptive mesh re-meshing technology is used for local refinement in the crack tip region. In this embodiment, eight-node quadrilateral reduced integral elements are used, and degenerate singular elements are used in the singular region at the crack tip to accurately capture the stress singularity at the crack tip.

[0026] Step S22: Solve the J integral using the region integration method to obtain the energy release rate; In this embodiment, the Type I stress intensity factor of the structural component to be evaluated is denoted as KⅠ, and the Type II stress intensity factor is denoted as KⅡ.

[0027] In linear elastic fracture mechanics, the energy release rate G is consistent with J and satisfies formula (1): (1) Where J is the J-integral; The effective Young's modulus is given by G, where G is the energy release rate. Effective Young's modulus The Young's modulus and Poisson's ratio of the structural component to be evaluated are determined. These values ​​are provided in material handbooks or experimentally and will not be elaborated upon here.

[0028] Step S23: Based on the fatigue crack propagation finite element simulation model and the energy release rate, determine the Type I stress intensity factor and Type II stress intensity factor of the structural component to be evaluated through interaction integration; In this embodiment, the interaction integral is calculated by superimposing the actual finite element solution with the auxiliary asymptotic field corresponding to a pure Type I load or a pure Type II load, denoted as . ; Furthermore, by sequentially selecting auxiliary fields as the unit mode I field and the unit mode II field, the corresponding interaction integrals can be obtained. and This is used to decouple and extract the Type I stress intensity factor KⅠ and Type II stress intensity factor KⅡ of the hybrid mode, and their expressions are shown in formula (2): (2) It is understandable that formula (1) can only yield the overall contribution of the mixed modes, and therefore cannot identify the two modal components separately. In order to distinguish these contributions, this invention solves the two types of stress intensity factors by calculating the interaction integral. For any given crack geometry and load level, it provides the mixed mode stress intensity factor required for subsequent crack propagation updates.

[0029] Step S24: Determine the crack propagation direction based on the Type I stress intensity factor and the Type II stress intensity factor; Specifically, in step S24, the circumferential stress is determined based on the maximum circumferential stress criterion, and the crack propagation direction is determined based on the circumferential stress, the Type I stress intensity factor, and the Type II stress intensity factor.

[0030] In this embodiment, the crack direction is determined using the maximum circumferential stress criterion. The circumferential stress is determined by the dominant term of the near-tip elastic field, expressed in polar coordinates (r, θ), centered on the crack tip. .

[0031] Furthermore, the specific formula for calculating the crack propagation angle is shown in formula (3): (3) in, The crack propagation angle; In this embodiment, formula (3) may have multiple roots in the interval [-π, π]. An acceptable branch is defined as maximizing the circumferential stress. And crack propagation angle The sign of the branch opposite to that of the Type II stress intensity factor KII is chosen to determine the direction associated with local tensile propagation ahead of the crack tip and to be consistently applied during the incremental construction of the crack path.

[0032] Step S25: Calculate the equivalent stress intensity factor range based on the Type I stress intensity factor and the Type II stress intensity factor; obtain the deterministic fatigue life function based on the equivalent stress intensity factor range.

[0033] In this embodiment, the calculation formula for the range of the equivalent stress intensity factor is specifically shown in formula (4): (4) in, The range of equivalent stress intensity factor, and These represent the value ranges of Type I stress intensity factor KⅠ and Type II stress intensity factor KⅡ within one stress cycle, respectively.

[0034] Furthermore, based on the Paris-Erdogan law and the aforementioned equivalent stress intensity factor range... The crack propagation rate is determined as shown in formula (5): (5) Where da / dN is the crack propagation rate, representing the increment of crack length per load cycle; a is the crack size; and N is the number of load cycles.

[0035] Due to the range of equivalent stress intensity factor As the crack configuration and local propagation direction evolve, fatigue life is obtained through incremental accumulation, rather than through closed-form integral calculation. In the i-th propagation step, the crack tip propagates along the direction... As the object moves, its geometric parameters are updated accordingly.

[0036] Furthermore, the cyclic increment is shown in formula (6): (6) in, For cyclic increment, This represents the incremental size of the crack.

[0037] Specifically, in step S25, the total fatigue crack propagation life in the deterministic fatigue life function is determined by adding the cyclic increments of the crack propagation history.

[0038] In this embodiment, the total fatigue crack propagation life is as shown in formula (7): (7) in, The total fatigue crack propagation life; The total number of propagation steps required for a crack to extend from its initial length to a specified critical length.

[0039] Furthermore, a deterministic mapping relationship from input variables to fatigue life can be defined, i.e., a deterministic fatigue life function, as shown in formula (8): (8) in, It is a deterministic fatigue life function; This represents an uncertain input parameter vector, where n is the total number of uncertain input parameter vectors; [·] T This represents the transpose of a vector.

[0040] Step S3: Determine the independent variable function with respect to the uniformly distributed variable vector based on the deterministic fatigue life function, where: Based on the judgment result that there is a statistical correlation between the uncertain input parameter vectors, the Rosenblatt transformation is used to map the statistically correlated uncertain input parameter vectors into statistically independent uniformly distributed variable vectors. Based on the judgment result of statistical independence between the uncertain input parameter vectors, the uniformly distributed variable vectors that are statistically independent of each other are determined according to the uncertain input parameter vectors. In this embodiment, if there is a statistical correlation between the uncertain input parameter vectors, the Rosenblatt transformation is used to map the statistically correlated uncertain input parameter vectors to the statistically independent uniformly distributed variable vectors, wherein the forward transformation is as shown in formula (9): (9) in When given a preceding variable The conditional cumulative distribution function.

[0041] Under formula (9), the transformed vector It consists of independent uniform variables on the interval [0,1].

[0042] The inverse transformation is shown in Equation (10): (10) Furthermore, let T denote the inverse Rosenblatt mapping defined by equation (10).

[0043] The response of the random variable to crack propagation life is shown in Equation (11): (11) in, Let be an independent variable function of a uniformly distributed variable vector; Y is a random variable representing crack propagation life.

[0044] If the uncertain input parameter vectors are statistically independent, then the response of the crack propagation life random variable is as shown in formula (12): (12) It is understandable that fatigue life is a scalar response, that is... The challenge lies in its dual nature. First, deterministic models are computationally expensive, making repeated direct sampling or multidimensional integration in physical space costly. Second, the joint density function does not require factorization because the components of X may exhibit statistical correlations. In fatigue crack propagation applications, this problem is particularly critical for parameters such as C and M, which are typically extracted from the same experimental data and therefore cannot be naturally modeled as independent variables. This invention addresses the challenge of accurately describing the true correlation of material parameters such as C and M in engineering using traditional decoupling methods through Rosenblatt transformation, which is independent of specific distribution types, and handles arbitrary marginal distributions and nonlinearly correlated structures, thereby improving the physical realism of the model.

[0045] Step S4: Decompose the independent variable function using the multiplicative dimensionality reduction method, and calculate the fractional moments of the lifetime response based on Gaussian quadrature. Specifically, step S4 includes: Step S41: Select a reference point in the space of independent variables; In this embodiment, the reference point is selected as the midpoint, and the reference point in the independent variable space is represented as shown in formula (13): (13) Step S42: Use multiplicative dimensionality reduction approximation to decompose the independent variable function into a product of multiple univariate functions; In this embodiment, for the case where the uncertain input parameter vectors are statistically independent, the fractional moments are as shown in formula (14): (14) in, It represents the fractional order.

[0046] For the case of statistical correlation between the uncertain input parameter vectors, the fractional moments are given by formula (15): (15) Furthermore, the multiplicative dimensionality reduction method is used to overcome its limitations. The sum of the single-variable cutoff values ​​at each point is used as an approximation, and the exponential operation yields formula (16): (16) Substituting the fractional moment (14), we get formula (17): (17) For cases where the statistical correlation between input parameter vectors is uncertain, since the transformed variables are now independent and uniformly distributed, the same first-order multiplication dimensionality reduction method used in the independent case can be directly applied. Based on the same reference point defined in formula (13). Substituting the fractional moment (15), we get formula (18): (18) Step S43: Select a set of fractional orders and calculate the fractional moments of the lifetime response based on Gaussian quadrature.

[0047] Specifically, in step S43, each fraction order is a non-integer.

[0048] In this embodiment, each one-dimensional integral in formulas (17) and (18) is calculated again in the interval [0,1] using the Gauss-Legend quadrature method. Taking formula (18) as an example, formula (19) can be obtained: (19) in, and These are the Gaussian-Legendary abscissa and weight mapped to the interval [0,1], respectively; j=1,2,3,...,N q .

[0049] It is understandable that if each dimension were directly given... When the tensor product integral method is applied to fractional moments (14) and (15), it is necessary to perform... Sub-deterministic model evaluation. In contrast, Gaussian quadrature only requires... The evaluation is performed once at the reference point and then along each univariate slice. This invention reduces multidimensional integrals to one-dimensional integrals using a multiplicative dimensionality reduction method, decreasing the number of FEA calls from 10 in the Monte Carlo method. 5 ~10 6 The number of iterations is reduced to 20-40, improving computational efficiency by 3-4 orders of magnitude. This reduces computational tasks that originally took weeks to just hours or even tens of minutes, enabling high-fidelity finite element models to directly participate in probabilistic design.

[0050] Step S5: Construct an optimization model based on the constraints of the fractional moments as the maximum entropy principle, solve for the Lagrange multipliers and fractional order, and obtain the probability density function of the fatigue life of the structural component to be evaluated.

[0051] Specifically, step S5 includes: Step S51: Using the normalization condition and the fractional moments as constraints, construct a maximum entropy optimization model with the goal of maximizing Shannon entropy. Specifically, the support set of the crack propagation life random variable is Ω = (0, ∞).

[0052] In this embodiment, Y is a random variable representing crack propagation life, and its support set is Ω = (0, ∞). For any acceptable probability density function... The Shannon entropy is shown in formula (20): (20) Furthermore, let α1, α2, ..., α m Given a sequence of real numbers, the corresponding moments... exist.

[0053] The reconstruction problem is expressed as formula (21): (twenty one) Furthermore, the feasible set defined by formula (21) is a convex set, and the entropy functional in formula (20) is strictly concave on this set. Therefore, as long as the moment constraint is feasible and the resulting density is normalizable, the solution is unique. According to the standard variational proof method, the maximization density function has an exponential form, as shown in formula (22): (twenty two) Where λ0, λ1, ..., λ m These are Lagrange multiplier constraints related to normalization and moments; Furthermore, a partition function is introduced, as shown in formula (23): (twenty three) Therefore, the normalization condition is given, as shown in formula (24): (twenty four) in, This is derived from the normalization condition of the probability density function. It's understandable that fractional moments are used instead of integer moments as constraints to maximize the entropy function because the estimation error of integer moments increases significantly with increasing order, leading to a larger error in the estimated probability density distribution. Compared to integer moments, fractional moments effectively solve this problem; applying fractional moments maximizes the use of existing information to estimate the desired distribution.

[0054] Step S52: Construct the Kullback-Leibler divergence between the true probability density function of fatigue life and the estimated probability density function of fatigue life; In this embodiment, the true probability density function of fatigue life and the estimated probability density function The distance between them is represented by the Kullback-Leibler divergence, as shown in Equation (25): (25) Furthermore, formula (21) Values ​​and formula (22) Substituting the values ​​into formula (25), we get formula (26): (26) Understandably, in the expression In this case, it is not necessary to know the parameters. The values ​​of these values ​​are used as prior information, and they can be obtained through subsequent optimization procedures; Understandably, in probability theory and information theory, Kullback-Leibler divergence, or KL distance for short, is used to measure the distance between two probability distribution functions. KL distance represents the similarity between two distributions; the smaller the Kullback-Leibler value, the closer the two distributions are. Therefore, to obtain the optimal fatigue life probability density function that maximizes entropy... and the estimated probability density function The distance between them can be represented by the Kullback-Leibler divergence.

[0055] Step S53: Solve for the minimized Lagrange multipliers and fractional order to minimize the Kullback-Leibler divergence; In step S53, the optimization procedure for minimizing the Lagrange multipliers and the fractional order is as follows: ; Where I is the objective function; Y is the crack propagation life random variable; and y is the realization of the crack propagation life random variable Y. It is a fractional moment; For Lagrange multipliers; Let be the fractional order; m is the number of fractional moments, i = 1, 2, ..., m.

[0056] It is understandable that the actual distribution of fatigue crack propagation is independent of... and Therefore, minimizing the KL distance means minimizing the last two terms in the KL expression, which gives formula (27): (27) Therefore, the distribution parameter that maximizes entropy, i.e., the Lagrange multiplier, is... and fractional order This can be determined through program optimization.

[0057] Step S54: Determine the probability density function of the fatigue life of the structural component to be evaluated based on the Lagrange multipliers.

[0058] In this embodiment, the probability density function of fatigue life is shown in formula (28): (28) To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and specific examples. Examples 1 and 2 correspond to typical aerospace structures or standard test pieces of different complexities, and all use Monte Carlo simulations as comparative examples. Example 1

[0059] Example 1 uses a standard specimen with a central crack as the analytical benchmark to verify the accuracy and efficiency of the method of the present invention in reconstructing the fatigue life probability distribution when there is statistical correlation in the input parameters. The deterministic fatigue life model of Example 1 is established based on the Paris-Erdogan formula, with the initial crack size, load amplitude, and geometric factor taken as constant values. The uncertain input parameter vector consists of the Paris coefficient C and the Paris exponent M, both of which follow a log-normal distribution. The distribution parameters are calibrated using fatigue test data of typical aluminum alloy materials. Based on the joint fitting results of the same set of crack propagation tests, a statistical correlation is identified between C and M, with the correlation coefficient in the benchmark case taken as a moderately strong positive value.

[0060] according to Figure 1 The flowchart shown in this embodiment of the invention considers a high-efficiency and high-precision assessment method for crack propagation life of aerospace structures with parameter correlation. Step S1 determines that there is a statistical correlation between C and M. Step S2 directly establishes a deterministic function from the input parameters to the fatigue life using analytical expressions. Step S3 maps the correlated C and M to independent uniformly distributed variables through Rosenblatt transformation, obtaining the life function with respect to the independent variables. Step S4 selects the median as the reference point, uses the multiplicative dimensionality reduction method to decompose the life function into the product of two univariate functions, and performs a 5th-order Gauss-Legendal integral on each univariate cross-section, performing a total of 11 deterministic life assessments to calculate the fractional moments. Step S5 uses the obtained fractional moments as constraints to reconstruct the probability density function of fatigue life through maximum entropy optimization. The comparison uses the same joint distribution for 10... 6 Monte Carlo sampling, totaling 10 6 For each sample, a reference histogram and survival probability curve are obtained.

[0061] The resulting probability density function is, for example... Figure 2 As shown, this is a comparison of the probability density function of the fatigue life of the central crack plate in the efficient and high-precision evaluation method for the crack propagation life of aerospace structures considering parameter correlation in an embodiment of the present invention. In the figure, M-DRM is the probability density curve reconstructed by the multiplicative dimensionality reduction method, and MCS is the histogram obtained by Monte Carlo simulation. As can be seen from the figure, when the correlation coefficient ρ=0.4 and the limiting case ρ=1.0, the probability density curve reconstructed by the multiplicative dimensionality reduction method of the present invention is highly consistent with the Monte Carlo histogram in terms of principal mode, dispersion, and left and right tails. This shows that the method of the present invention, which combines the transformation space M-DRM with the fractional moment maximum entropy reconstruction, can achieve accurate restoration at the level of the complete probability density function.

[0062] Survival probability comparison Figure 3As shown, this is a comparison of the survival probability curves of the central crack plate in the efficient and high-precision evaluation method for the crack propagation life of aerospace structures considering parameter-related factors according to an embodiment of the present invention. In the figure, the left and right parts are respectively at two preset life thresholds (i.e., 3×10). 4 and 5×10 4 The survival probability varies with the number of iterations (in the next iteration); M-DRM represents the survival probability reconstructed using multiplicative dimensionality reduction, and MCS represents the survival probability obtained using Monte Carlo simulation. As can be seen from the figure, at correlation coefficients ρ=0.4 and the limiting case ρ=1.0, the survival probability varies with the number of iterations (in the next iteration). -4 The relative error at the survival probability level is consistently below 5%, indicating that the reconstruction method of this invention can not only accurately fit the main distribution, but also effectively preserve the tail features related to reliability. For low-probability assessment scenarios, the accuracy of the tail behavior restoration of this method has met the requirements.

[0063] Impact analysis of correlation coefficient, such as Figure 4 As shown, this is a comparison of the survival probability of different preset lifetime thresholds when considering the correlation coefficient changes of the parameter-dependent high-efficiency and high-precision evaluation method for the crack propagation lifetime of aerospace structures in an embodiment of the present invention. In the figure, the left and right parts are respectively at two preset lifetime thresholds (i.e., 3×10). 4 and 5×10 4 The graph shows the survival probability under the following cycle as a function of the correlation coefficient ρ, where the correlation coefficient ρ ranges from -1 to 1. As can be seen from the graph, at higher thresholds, when ρ = -1.0, the survival probability is approximately 2.6 times higher. -4 When ρ=1.0, it increases to approximately 9.1 times. -3 This indicates that, within the permissible range of correlation, the predicted survival probability can vary by more than an order of magnitude. This suggests that when C and M are statistically correlated, the shape of the fatigue life distribution, especially its upper tail, may undergo substantial changes. Therefore, the independence assumption may lead to non-conservative or overly conservative results in reliability estimation.

[0064] In summary, Example 1 demonstrates that the Rosenblatt transform can accurately reproduce a given relevant structure in the non-degenerate case and is stable under the endpoint limiting condition, accurately capturing the complete long-tailed lifetime distribution; the transform space M-DRM model only requires 11 deterministic model evaluations to obtain accurate fractional moments, improving computational efficiency by several orders of magnitude compared to the Monte Carlo method; and the maximum entropy reconstruction method can faithfully restore fatigue lifetime density and related survival probabilities.

[0065] Example 2 This embodiment uses an aluminum support with an angular crack subjected to combined Type I and Type II cyclic loads as an example to verify the effectiveness and computational benefits of the method of the present invention in actual finite element structures. In Example 2, the support base is fixed and subjected to a periodic uniform pressure with a load ratio of 0. The material model uses an isotropic linear elastic body with Young's modulus E = 30000 ksi and Poisson's ratio ν = 0.3. It is assumed that an angular crack with an initial length of a0 exists in the critical stress concentration region. The uncertain input parameters are X = [a0, P...]. max [C, M] T The probability characteristics are shown in Table 1. Table 1. Probabilistic characteristics of input variables for aluminum support specimens with corner cracks. ; Among them, a0 and P max The variables follow a normal distribution with a coefficient of variation of 0.10; C and M follow a log-normal distribution with a coefficient of variation of 0.30. Example 2 first analyzes the case where each variable is statistically independent, and then introduces statistical correlation between C and M separately for comparison.

[0066] First, a finite element simulation model of fatigue crack propagation in the support was established. The crack tip region was represented by eight-node quadrilateral reduced integral elements with adaptive refinement to ensure accurate analysis of steep stress gradients and singular fields near the crack tip. Figure 5 The image shown is a refined finite element mesh of an aluminum support with a corner crack, based on an embodiment of the present invention that considers parameter-dependent high-efficiency and high-precision evaluation method for crack propagation life of aerospace structures. The initial mesh contains 1674 elements and 3680 nodes. Crack propagation is simulated using an incremental method with a constant propagation step size of 0.075 inches. The analysis terminates when the crack reaches a critical length of 0.5 inches. The number of cycles is recorded as the fatigue life N. f .

[0067] The stress intensity factor obtained from the finite element method is compared with that obtained from the FRANC2D software along the path, for example. Figure 6 As shown in the figure, the embodiment of the present invention considers the variation of the stress intensity factor of the support along the crack propagation path in the efficient and high-precision evaluation method of crack propagation life of aerospace structures with parameter correlation. In the figure, the left and right parts respectively compare the stress intensity factors of type I and type II obtained based on the finite element method (ABAQUS) with the corresponding results generated by FRANC2D software throughout the crack propagation process. As can be seen from the figure, the prediction results of the two modes are always highly consistent, and the relative deviation is always kept within 1.5% throughout the propagation process. This shows that the mesh density, boundary conditions, constitutive assumptions and crack tip treatment methods adopted in Example 2 fully meet the current probabilistic analysis requirements.

[0068] according to Figure 1The flowchart shown in this embodiment of the invention considers a high-efficiency and high-precision assessment method for the crack propagation life of aerospace structures with parameter correlation. Step S1 determines the correlation structure between parameters. Step S2 establishes a finite element simulation model of fatigue crack propagation of the support, extracts Type I and Type II stress intensity factors through J-integral and interaction integral, determines the crack propagation direction according to the maximum circumferential stress criterion, and obtains the deterministic fatigue life by iteratively accumulating the equivalent stress intensity factor range along the propagation path. Step S3 obtains independent uniform variables using marginal probability integral transformation in the independent case, and transforms the correlation relationship to an independent space using Rosenblatt transformation in the correlated case. Step S4 performs first-order multiplication dimensionality reduction with the midpoint as the reference point, assigning 5 Gaussian integration points to each variable direction, requiring only 21 finite element evaluations to calculate the fractional moments. Step S5 performs maximum entropy reconstruction based on the fractional moments to obtain the fatigue life probability density function.

[0069] Example 2 uses 10 5 The Monte Carlo simulation is used as a reference solution.

[0070] Lifetime probability density pair, for example Figure 7 As shown, this is a comparison of the fatigue life probability density function of the support structure in the efficient and high-precision evaluation method for crack propagation life of aerospace structures considering parameter correlation according to an embodiment of the present invention. In the figure, M-DRM is the maximum entropy reconstruction result obtained based on 21 deterministic simulations, and MCS is based on 10 5 The reference Monte Carlo simulation results for a sample are shown in the figure, and the reconstruction density matches the Monte Carlo histogram well.

[0071] Furthermore, the average fatigue life calculated by the method of the present invention is 2.9689 × 10⁻⁶. 4 In the next cycle, the Monte Carlo reference value was 2.9348 × 10⁻⁶. 4 In the second cycle, the relative error of the mean was only 1.16%; the relative difference of the standard deviation was 3.14%. This shows that even using the first-order form, the method of the present invention can still provide sufficiently high-quality moment estimates and achieve accurate density reconstruction.

[0072] Furthermore, the direct MCS method requires 10... 5 The proposed method requires only 21 evaluations of the unconditional distribution, achieving approximately 4.76 × 10⁻⁶ crack propagation analysis, whereas the proposed method only requires 21 evaluations of the unconditional distribution. 3 The computational workload is reduced by a factor of two.

[0073] The effects of parameter correlation, such as Figure 8 and Figure 9As shown, these are respectively the fatigue life probability density function of the support under different correlation coefficients of the efficient and high-precision evaluation method for the crack propagation life of aerospace structures considering parameter correlation in the embodiments of the present invention, and the survival probability curve of the support under different correlation coefficients of the efficient and high-precision evaluation method for the crack propagation life of aerospace structures considering parameter correlation in the embodiments of the present invention. Specifically; Figure 8 The figure shows the fatigue life density reconstruction results for three cases: ρ=-1, ρ=0, and ρ=1. As can be seen, positive correlation shifts the distribution towards longer fatigue lives and forms a significantly thickened upper tail distribution; while negative correlation leads to increased distribution concentration and decreased upper tail quality. Figure 9 The survival probability curves are shown for three scenarios: ρ=-1, ρ=0, and ρ=1. As can be seen from the figure, the elimination of this correlation is negligible in both magnitude and spatial distribution. Meanwhile, at a lifetime threshold of 5×10... 5 In the next iteration, the survival probability varies by more than an order of magnitude within the range of the examined correlation. This indicates that, for this type of interval problem, the correlation structure of the Paris parameter has a direct and practically significant impact on the reliability-related tail of the fatigue life distribution.

[0074] In summary, this invention can achieve high-precision full-lifetime probabilistic reconstruction with very few model calls in complex finite element crack propagation analysis, and correctly preserve the statistical correlation effect between parameters.

Claims

1. A highly efficient and accurate method for evaluating the crack propagation life of aerospace structures considering parameter-related factors, characterized in that, include: Step S1: Define the uncertain input parameter vector that affects the fatigue life of the structural component to be evaluated, and determine the statistical correlation between the uncertain input parameter vectors. Step S2: Based on finite element analysis, establish a finite element simulation model of fatigue crack propagation of the structural component to be evaluated under the combined action of type I and type II loads. Determine the deterministic fatigue life function of the structural component to be evaluated based on the fatigue crack propagation finite element simulation model and the uncertain input parameters. Step S3: Determine the independent variable function with respect to the uniformly distributed variable vector based on the deterministic fatigue life function, where: Based on the judgment result that there is a statistical correlation between the uncertain input parameter vectors, the Rosenblatt transformation is used to map the statistically correlated uncertain input parameter vectors into statistically independent uniformly distributed variable vectors. Based on the judgment result of statistical independence between the uncertain input parameter vectors, the uniformly distributed variable vectors that are statistically independent of each other are determined according to the uncertain input parameter vectors. Step S4: Decompose the independent variable function using the multiplicative dimensionality reduction method, and calculate the fractional moments of the lifetime response based on Gaussian quadrature. Step S5: Construct an optimization model based on the constraints of the fractional moments as the maximum entropy principle, solve for the Lagrange multipliers and fractional order, and obtain the probability density function of the fatigue life of the structural component to be evaluated.

2. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 1, characterized in that, Step S2 includes: Step S21: Based on finite element analysis, establish a finite element simulation model of fatigue crack propagation of the structural component to be evaluated under the combined action of type I and type II loads; Step S22: Solve the J integral using the region integration method to obtain the energy release rate; Step S23: Based on the fatigue crack propagation finite element simulation model and the energy release rate, determine the Type I stress intensity factor and Type II stress intensity factor of the structural component to be evaluated through interaction integration; Step S24: Determine the crack propagation direction based on the Type I stress intensity factor and the Type II stress intensity factor; Step S25: Calculate the equivalent stress intensity factor range based on the Type I stress intensity factor and the Type II stress intensity factor; obtain the deterministic fatigue life function based on the equivalent stress intensity factor range.

3. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 2, characterized in that, In step S24, the circumferential stress is determined based on the maximum circumferential stress criterion, and the crack propagation direction is determined based on the circumferential stress, the Type I stress intensity factor, and the Type II stress intensity factor.

4. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 3, characterized in that, In step S25, the total fatigue crack propagation life in the deterministic fatigue life function is determined by adding the cyclic increments of the crack propagation history.

5. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 1, characterized in that, The uncertain input parameter vector includes the initial crack size, load amplitude, Paris coefficient, and Paris exponent.

6. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 5, characterized in that, Step S4 includes: Step S41: Select a reference point in the space of independent variables; Step S42: Use multiplicative dimensionality reduction approximation to decompose the independent variable function into a product of multiple univariate functions; Step S43: Select a set of fractional orders and calculate the fractional moments of the lifetime response based on Gaussian quadrature.

7. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 6, characterized in that, In step S43, the order of each fraction is a non-integer.

8. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 7, characterized in that, Step S5 includes: Step S51: Using the normalization condition and the fractional moments as constraints, construct a maximum entropy optimization model with the goal of maximizing Shannon entropy. Step S52: Construct the Kullback-Leibler divergence between the true probability density function of fatigue life and the estimated probability density function of fatigue life; Step S53: Solve for the minimized Lagrange multipliers and fractional orders to minimize the Kullback-Leibler divergence; Step S54: Determine the probability density function of the fatigue life of the structural component to be evaluated based on the Lagrange multipliers.

9. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 8, characterized in that, In step S53, the optimization procedure for minimizing the Lagrange multipliers and the fractional order is as follows: ； Where I is the objective function; Y is the crack propagation life random variable; and y is the realization of the crack propagation life random variable Y. It is a fractional moment; For Lagrange multipliers; For fractional order; Let i be the number of fractional moments, i = 1, 2, ..., m.

10. The efficient and high-precision assessment method for crack propagation life of aerospace structures considering parameter correlation as described in claim 9, characterized in that, The support set of the random variable Y representing crack propagation life is Ω = (0, ∞).

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