An uncertainty quantification and structural reliability optimization method for insufficient data

Abstract

The application provides an uncertainty quantification and structural reliability optimization method for insufficient data, and relates to the field of reliability optimization design of aerospace main load-bearing structures. The application simulates various possible conditions under insufficient data / small sample data conditions by repeatedly resampling the known data, further constructs an extended ellipsoid model based on bootstrap to envelope the various possible conditions, realizes high-precision uncertainty quantification under insufficient data in a manner of appropriately expanding the ellipsoid volume, further utilizes an enhanced step adjustment method (ASSA) to carry out non-probabilistic reliability optimization design under insufficient data and obtains a required reliable result. The application solves the problem that a traditional non-probabilistic ellipsoid model cannot accurately quantify the uncertainty boundary under insufficient data conditions, and further leads to invalidation of the reliability optimization result, and is expected to become one of the most potential technical means for reliability optimization design problems involving insufficient samples in the engineering field.

Description

Technical Field

[0001] This invention relates to the field of reliability optimization design of aerospace main load-bearing structures, and in particular to a method for uncertainty quantification and structural reliability optimization oriented towards insufficient data. Background Technology

[0002] Uncertainty objectively exists in all stages of engineering structure production, manufacturing, and service, exhibiting characteristics such as wide sources (material deviations, manufacturing errors, load deviations, etc.) and strong randomness (uncertainty in location and distribution). Traditional deterministic design methods based on safety factors are insufficient to characterize the propagation laws of multi-source uncertainties, easily leading to excessive redundancy in structural design. Although structural reliability analysis methods can realize the propagation of input uncertainties to structural response uncertainties, the accuracy of their results heavily depends on the precision of the uncertainty quantification model.

[0003] Due to the high manufacturing and testing costs, especially for large aerospace structures, the amount of sample data available in practice is usually very limited. This makes it difficult to construct accurate uncertainty quantification models, leading to significant biases in structural failure risk assessments. Compared to probabilistic models, non-probabilistic models can alleviate the need for a huge number of samples to some extent by constructing boundaries for uncertainty parameters. However, when faced with insufficient data, traditional non-probabilistic modeling strategies based on minimum volume can result in an overly compact uncertainty domain, thus exposing the structural design to a greater risk of failure. Therefore, there is an urgent need to develop a high-precision uncertainty quantification and reliability optimization method for insufficient data. Summary of the Invention

[0004] To address the problem that traditional non-probabilistic ellipsoidal models struggle to accurately quantify uncertainty boundaries under insufficient data conditions, leading to the failure of reliability optimization results, this invention proposes a bootstrap-based extended ellipsoid model (BEEM) and develops a non-probabilistic reliability optimization design method for insufficient data. This method simulates various possible scenarios under insufficient / small sample data conditions by repeatedly resampling known data. It further constructs a bootstrap-based extended ellipsoidal model to enclose these various possible scenarios, achieving high-precision uncertainty quantification under insufficient data by appropriately increasing the ellipsoid volume. Finally, it utilizes the augmented step size adjustment (ASSA) method to conduct non-probabilistic reliability analysis under insufficient data, ultimately obtaining reliable results that meet design requirements.

[0005] The objective of this invention is mainly achieved through the following technical solutions:

[0006] A method for uncertainty quantification and structural reliability optimization for inadequate data includes the following steps:

[0007] S1. Based on the reliability optimization design requirements of aerospace structures, define a mathematical model for reliability optimization, including determining the objective function, constraint function, design variables, random variables, and setting the target reliability index and initial design point.

[0008] S2. For the mathematical optimization model obtained in step S1, perform non-probabilistic uncertainty quantification under insufficient data, collect relevant data of random variables to form an initial dataset, and calculate the sample mean and covariance of the initial dataset; perform matrix eigenvalue analysis on the sample covariance of the initial dataset to obtain the eigenvector matrix, which is the direction of the semi-major axis of the initial ellipsoid; set the total number of bootstrap samplings to Nsim; set the upper limit of one-sided confidence α; set the number of samplings kk = 1.

[0009] S3. Use Bootstrap to perform a resampling to obtain the resampled dataset. Calculate the sample mean and covariance of the resampled dataset, and construct the Bootstrap ellipsoid model based on the gradient optimization algorithm.

[0010] S4. Approximately uniformly discretize the boundary of the above bootstrap ellipsoid model to obtain a set of boundary points. Construct a boundary point ellipsoid model with the direction of the semi-major axis of the initial ellipsoid as the given direction. Solve for the optimal semi-major axis length of the boundary point ellipsoid model based on the gradient optimization algorithm.

[0011] S5. Let kk = kk + 1, and repeat the above steps S3-S4 until kk = Nsim, to obtain the optimal semi-major axis length of the boundary point ellipsoid for each sampling.

[0012] S6. Statistically determine the optimal ellipsoidal semi-major axis length distribution of the boundary point ellipsoidal model, eliminate extreme cases using an outlier filtering mechanism, and further obtain the ellipsoidal semi-major axis length at a specified confidence level using the upper limit of one-sided confidence and construct an extended ellipsoidal model.

[0013] S7. To facilitate subsequent nonprobabilistic reliability optimization, the random variables obtained in step S1 are transformed into the standard q-space based on the extended ellipsoid model.

[0014] S8. Based on the extended ellipsoid model, define the structural nonprobabilistic reliability optimization formula (double-layer nested optimization); carry out nonprobabilistic reliability optimization based on the initial design point set in step S1.

[0015] S9. Based on the current design point, use the first-order second-moment algorithm to solve the inner-layer reliability analysis in non-probabilistic reliability optimization and obtain the performance points of interest.

[0016] S10. Based on the function value and gradient information at the performance points of interest, determine the new outer layer design points.

[0017] S11. Repeat S9-S10 until the convergence condition is met, and obtain the optimal design result with nonprobabilistic reliability.

[0018] Furthermore, in step S1:

[0019] The mathematical model for optimizing the structural reliability is shown in Equation (1);

[0020]

[0021] In the formula, f(·) is the objective function; d is the design variable; X is a random variable; g j (·) represents the j-th constraint / function; P f (·) represents the failure probability of the j-th functional function; m is the number of functional functions; d is the target reliability index for the j-th function; L and d U These are the upper and lower bounds of the design variables; the initial design point is set to d. 0 ;

[0022] The objective function includes, but is not limited to, structural mass or overall structural stiffness; the constraint function is the failure probability of the structural mechanical response, including the static, dynamic, or buckling response characteristics of the structure.

[0023] Furthermore, step S2 specifically includes the following:

[0024] Collect relevant data of random variable X to form an initial dataset x = [x 1 ,x 2 ,…,x ns ]; where ns is the number of initial data;

[0025] Calculate the mean of the initial data As shown in formula (2);

[0026]

[0027] In the formula, Let be the l-th set of data for the i-th design variable, and n be the number of design variables;

[0028] Calculate the initial data covariance matrix C 0 As shown in formula (3);

[0029]

[0030] In the formula, C ij C0 The element at the corresponding position;

[0031] For C 0 Perform eigenvalue decomposition to calculate the direction of the semi-major axis of the initial ellipsoid. As shown in formula (4);

[0032]

[0033] In the formula, λ 0 It is an eigenvalue matrix.

[0034] Furthermore, in steps S3 and S4, constructing the bootstrap ellipsoid and the boundary point ellipsoid requires solving the optimization formula using gradient-based optimization algorithms, including but not limited to the interior point method, the effective set method, or the sequential quadratic programming method.

[0035] Furthermore, in step S3, resampling is performed using bootstrap, i.e., sampling with replacement in the initial dataset while keeping the data volume unchanged, to obtain the kk=1th bootstrap sample dataset.

[0036] The kk=1th Bootstrap sample dataset mean Covariance It can also be calculated using formulas (2) and (3) respectively;

[0037] With the length of the semi-major axis of the ellipsoid As the design variable, minimizing the ellipsoidal volume is the objective function. An optimization formula is constructed and solved for the given mean. and the direction of the semi-major axis of the ellipsoid The minimum volume ellipsoid of the lower envelope initial data is obtained as the bootstrap ellipsoid model, as shown in formula (5);

[0038]

[0039] In the formula, The bootstrap ellipsoid feature matrix is ​​given by the k=1th sampling; the direction of the semi-major axis of the ellipsoid. It can be derived from the covariance matrix Calculated using formula (4); This is the inverse of the determinant of the bootstrap ellipsoid characteristic matrix, which, since it is proportional to the volume of the ellipsoid, minimizes... Equivalent to minimizing the volume of the ellipsoid; x i This represents the i-th initial data; This represents the mean of the k=1th bootstrap sample dataset;

[0040] The semi-major axis length of the optimal bootstrap ellipsoid is obtained by solving equation (5) using the sequential quadratic programming method in Matlab.

[0041] Construct the bootstrap ellipsoid model, and its boundary equation is shown in Equation (6);

[0042]

[0043] In the formula, the optimal characteristic matrix of the bootstrap ellipsoid under the k=1th sampling is... As shown in formula (7);

[0044]

[0045] Furthermore, in step S4, a set of boundary points is obtained by approximately uniform sampling on the boundary equation (6). Where n bp The number of boundary points;

[0046] With the length of the semi-major axis of the ellipsoid As the design variable, minimizing the volume of the boundary point ellipsoid is the objective function. An optimization formula is constructed and solved for the given mean. and the direction of the semi-major axis of the ellipsoid The minimum volume ellipsoid of the lower envelope boundary point data is obtained, i.e. the boundary point ellipsoid, as shown in formula (8);

[0047]

[0048] In the formula, This is the ellipsoidal feature matrix of the boundary point corresponding to the kk=1th sampling; The direction of the semi-major axis of the initial data ellipsoid can be calculated from (4); This is the inverse of the determinant of the characteristic matrix of the ellipsoid at the boundary points. Since it is proportional to the volume of the ellipsoid, minimizing it is possible. This is equivalent to minimizing the volume of the boundary point ellipsoid; This represents the i-th boundary point of the optimal boostrap ellipsoid obtained by the k=1th sampling.

[0049] The optimal semi-major axis length of the boundary point ellipsoid corresponding to the kk=1th sampling is obtained by solving formula (8) using the sequential quadratic programming method in Matlab.

[0050] Furthermore, in step S6, an outlier screening mechanism is used to exclude extreme cases, including the 3σ criterion, box plot method, cluster analysis, or kernel density estimation method. The 3σ criterion will be used as an example:

[0051] The optimal semi-major axis lengths of the ellipsoids at the Nism sub-boundary points are used to form a set of semi-major axis lengths.

[0052] calculate The mean of the i-th dimension and standard deviation The upper and lower bounds of the interval are calculated using the 3σ criterion.

[0053] against In the i-th dimension, elements within the upper and lower bounds are selected and sorted in descending order. The length of the semi-major axis of the ellipsoid corresponding to the upper limit of the one-sided confidence score α is calculated.

[0054] Traversal For all dimensions i = 1, 2, ..., n, we obtain the set of ellipsoidal semi-major axis lengths corresponding to the one-sided confidence upper limit α. Where n is the number of design variables;

[0055] Calculate the characteristic matrix W of the extended ellipsoid model BEEM As shown in formula (9);

[0056]

[0057] Construct an extended ellipsoid model, the boundary equation of which is shown in formula (10);

[0058] Γ BEEM ={X|(Xx) m ) T W BEEM (Xx m )=1} (10)

[0059] Furthermore, in step S7, the specific steps for transforming the random variable into the standard q-space are as follows:

[0060] For the characteristic matrix W of the extended ellipsoid model BEEM Perform eigenvalue decomposition, as shown in formula (11);

[0061]

[0062] In the formula, Φ BEEM The corresponding normalized eigenvector matrix; Λ BEEM This is the corresponding eigenvalue matrix;

[0063] Transform the random variable X into a vector q in the standard q-space, as shown in formula (12);

[0064]

[0065] Furthermore, in step S8, the structural nonprobabilistic reliability optimization based on the performance-of-concern method can be defined as formula (13), which is essentially a double-nested optimization problem. The outer layer minimizes the objective function, and the inner layer is a nonprobabilistic reliability analysis (which is also essentially an optimization problem). Further, based on the initial design point d set in step S1... 0 Conduct nonprobabilistic reliability optimization;

[0066]

[0067] In the formula, G j (·) corresponds to the j-th functional value in q-space in formula (1); α j =G j (d,q * ) corresponds to the performance value of the j-th function in formula (1), which is equivalent to the function value corresponding to the minimum performance target point in the function measurement method. It can be calculated by solving formula (14).

[0068]

[0069] In the formula, Let be the target nonprobabilistic reliability index for the j-th functional function.

[0070] Furthermore, in step S9, the first-order second-moment algorithm includes, but is not limited to, the enhanced step size adjustment method (ASSA). Taking ASSA as an example, the following explanation is provided:

[0071] Based on the current design point, the ASSA algorithm is used to conduct non-probabilistic reliability analysis; the number of iterations for the inner non-probabilistic reliability analysis is set to k = 1;

[0072] Define the oscillation identification parameter ξ for the k-th iteration. k As shown in formula (15);

[0073]

[0074] In the formula, q k For the k-th iteration point; ||q k ||for q k The model; For the k-th iteration point q k The direction of the normalized negative gradient of the function;

[0075] When ξ k If the value is greater than 0, update the next iteration point using the iterative format of formula (16). When ξ k <0, update the next iteration point using the iterative format of formula (17).

[0076]

[0077]

[0078] Let k = k + 1, and repeat formulas (15)(16)(17) until... Stop iteration. This refers to the performance points that are of concern.

[0079] Furthermore, in step S10, based on the function value and corresponding gradient information at the performance point of interest, a new outer design point is determined using the sequential quadratic programming method in Matlab.

[0080] Furthermore, the convergence condition in step S11 is: the iteration stops when the relative error between two consecutive iteration points does not exceed 1E-6, thus obtaining the optimal design result with non-probabilistic reliability.

[0081] The beneficial effects of this invention are:

[0082] This invention proposes a method for uncertainty quantification and structural reliability optimization for insufficient data. By introducing the concept of bootstrap resampling, it quantifies the cognitive uncertainty induced by insufficient samples. An extended ellipsoidal uncertainty domain can be adaptively established according to the initial data quantity, realizing high-precision quantification of multi-source uncertainty under insufficient data. Then, combined with the ASSA algorithm, reliable results that meet the design requirements can be obtained. Attached Figure Description

[0083] Figure 1 This is a flowchart of the method of the present invention;

[0084] Figure 2 The boundary of the bootstrap ellipsoid model under the kk=1th sampling;

[0085] Figure 3 These are the discrete boundary points of the bootstrap ellipsoid model under the kk=1th sampling.

[0086] Figure 4 This is a diagram showing the distribution of the optimal semi-major axis length of the ellipsoid model at the boundary points. Figure 4 (a) is a diagram showing the distribution of the semi-major axis length of the first-dimensional optimal ellipsoid of the boundary point ellipsoid model; Figure 4 (b) is a diagram showing the distribution of the semi-major axis length of the second-dimensional optimal ellipsoid of the boundary point ellipsoid model;

[0087] Figure 5 An extended ellipsoid model based on Bootstrap;

[0088] Figure 6This is a comparison chart showing the results of the bootstrap-based extended ellipsoidal model proposed in this invention and the ellipsoidal model constructed by the traditional MVE method. Detailed Implementation

[0089] The present invention will be further described in detail below with reference to the accompanying drawings and specific examples, and by comparison with the minimum-volume ellipsoid (MVE) method. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0090] like Figure 1 As shown, the present invention proposes a method for uncertainty quantification and structural reliability optimization for insufficient data, comprising the following steps:

[0091] S1. According to the reliability optimization design requirements of aerospace structures, a mathematical optimization model is defined, including determining the objective function and constraint function. In this embodiment, the objective function is the structural mass, and the constraint function is the failure probability of the structural mechanical response. At the same time, design variables and random variables are determined; target reliability index is set; and initial design point is set. In the embodiment provided by this invention, a non-probabilistic reliability optimization design of a classic stiffened shell is used as an example for detailed explanation, and its definition is shown in formula (18).

[0092]

[0093] In the formula, f(·) is the objective function, representing the mass of the stiffened tube structure; g j (·) represents the j-th constraint function / functional constraint, which respectively represents the maximum stress constraint, the first-order linear buckling load constraint, and the first-order vibration frequency constraint of the stiffened tube structure; according to aerospace structural design requirements, the target reliability index β is generally required. t ≥3.0, therefore, the target reliability index of the function in this embodiment is set to 3.0; two-dimensional random variable X i These represent the equivalent skin thickness and equivalent stiffener height of the stiffened tube structure, respectively. Based on experimental data, a two-dimensional random variable X is defined. i Follows the mean d i The standard deviation is 0.3, and the initial design point d is set. (0) =[5.0,5.0] T .

[0094] S2. Conduct non-probabilistic uncertainty quantification of random variables under insufficient data. To simulate actual insufficient data conditions, randomly select a set (ns = 10) of samples from the true distributions of equivalent skin thickness and equivalent rib height as initial known data x = [x 1 ,x 2 ,…,x ns As shown in Table 1;

[0095] Table 1 Initial Data

[0096] X (4.8053,5.0500) (5.3543,4.4104) (4.7725,4.6190) (4.6671,5.3526) No. 5 6 7 8 X (4.7463,5.6087) (4.8324,5.1811) (5.0535,5.5344) (4.8282,4.9175) No. 9 10 X (4.9409,5.5321) (5.1759,4.4405)

[0097] Based on the initial data, the mean x of the initial data is calculated using formula (19). m =[4.9177, 5.0646] T ;

[0098]

[0099] Based on the initial data, the covariance matrix of the initial data is calculated using formula (20).

[0100]

[0101] In the formula C ij C 0 The element at the corresponding position;

[0102] Based on the initial data, for C 0 Perform eigenvalue decomposition to calculate the direction of the semi-major axis of the initial ellipsoid. As shown in formula (21);

[0103]

[0104] In the formula It is the eigenvalue matrix.

[0105] Set the total number of bootstrap samples Nsim = 1000; set the current number of samples kk = 1; set the upper limit of one-sided confidence α = 95%.

[0106] S3. Resampling using Bootstrap involves sampling with replacement from the initial dataset while keeping the data size constant, resulting in the Bootstrap sample dataset after the k=1th sampling. As shown in Table 2;

[0107] Table 2 shows the bootstrap sample dataset under the k=1th sampling.

[0108] X (5.1759,4.4405) (4.8282,4.9175) (4.8324,5.1811) (4.6671,5.3526) No. 5 6 7 8 X (4.8324,5.1811) (4.9409,5.5321) (4.8053,5.0500) (5.0535,5.5344) No. 9 10 X (5.1759,4.4405) (5.0535,5.5344)

[0109] Using formulas (19) and (20), the bootstrap sample dataset under the k=1th sampling is calculated. Sample mean Covariance

[0110] With the length of the semi-major axis of the ellipsoid As the design variable, minimizing the ellipsoidal volume is the objective function. An optimization formula is constructed and solved for the given mean. and the direction of the semi-major axis of the ellipsoid The minimum volume ellipsoid of the lower envelope initial data is obtained as the bootstrap ellipsoid, as shown in formula (22);

[0111]

[0112] In the formula, The bootstrap ellipsoid feature matrix is ​​given by the kk=1th sampling. The inverse of the determinant of the bootstrap ellipsoid characteristic matrix; the direction of the semi-major axis of the ellipsoid. It can be derived from the covariance matrix The result is obtained by calculation using equation (21);

[0113] The optimal bootstrap ellipsoid semi-major axis length is obtained by solving equation (22) using the sequential quadratic programming method in Matlab.

[0114] Construct the bootstrap ellipsoid model, whose boundary equation (23) is as follows: Figure 2 As shown;

[0115]

[0116] In the formula, The optimal eigenma matrix of the bootstrap ellipsoid under the kk=1th sampling.

[0117] S4. Approximately uniform sampling is performed on the boundary equation (23) to obtain a set of (n bp =200) boundary points like Figure 3 As shown;

[0118] With the length of the semi-major axis of the ellipsoid As the design variable, minimizing the volume of the boundary point ellipsoid is the objective function. An optimization formula is constructed and solved for the given mean. and the direction of the semi-major axis of the ellipsoid The minimum volume ellipsoid of the lower envelope boundary point data is obtained, i.e. the boundary point ellipsoid, as shown in formula (24);

[0119]

[0120] In the formula, This is the ellipsoidal feature matrix of the boundary point corresponding to the kk=1th sampling; The inverse of the determinant of the characteristic matrix of the boundary point ellipsoid; The direction of the semi-major axis of the initial data ellipsoid;

[0121] The sequential quadratic programming method in Matlab is used to solve formula (24) to obtain the optimal semi-major axis length of the boundary point ellipsoid model corresponding to the kk=1th sampling.

[0122] S5. Let kk = kk + 1, and repeat steps S3-S4 until kk = Nsim, to obtain the optimal semi-major axis length of the boundary point ellipsoid model for each sampling.

[0123] S6. The optimal ellipsoidal semi-major axis lengths of the statistical Nim sub-boundary point ellipsoidal model are used to form a set of semi-major axis lengths. Its two dimensions and The distribution is as follows Figure 4 As shown;

[0124] calculate The mean of the i-th dimension and standard deviation The upper and lower bounds of the interval are calculated using the 3σ criterion.

[0125] against For the i-th dimension, filter out elements within the upper and lower bounds and sort them in descending order. Calculate the length of the semi-major axis of the ellipsoid corresponding to the upper limit of one-sided confidence.

[0126] Traversal For all dimensions i = 1, 2, the ellipsoidal semi-major axis length corresponding to the one-sided α = 95% confidence upper limit is obtained.

[0127] The characteristic matrix of the extended ellipsoid model is calculated using formula (25).

[0128]

[0129] Construct an extended ellipsoid model, whose boundary equation (26) is as follows: Figure 5 As shown;

[0130] Γ BEEM ={X|(Xx) m ) T W BEEM (Xx m)=1} (26)

[0131] S7. The characteristic matrix W of the extended ellipsoid model BEEM Perform eigenvalue decomposition, as shown in formula (27);

[0132]

[0133] In the formula, This is the corresponding normalized eigenvector matrix; This is the corresponding feature matrix;

[0134] Transform the random variable X into a vector q in the standard q-space, as shown in formula (28);

[0135]

[0136] S8. In this embodiment, the structural nonprobabilistic reliability optimization based on the concern performance method can be defined as formula (29);

[0137]

[0138] In the formula, G j (·) corresponds to the j-th functional in q-space in formula (18); α j =G j (d,q * The value of interest for the j-th function in formula (18) can be calculated by solving formula (30);

[0139]

[0140] In the formula, Let be the target nonprobabilistic reliability index for the j-th functional function.

[0141] S9, Based on the current design point d (0) =[5.0,5.0] T The ASSA algorithm is used to conduct nonprobabilistic reliability analysis; the number of iterations for the inner nonprobabilistic reliability analysis is set to k = 1.

[0142] Define the oscillation identification parameter ξ k As shown in formula (31);

[0143]

[0144] In the formula, q k For the k-th iteration point; ||q k ||for q k The model; For the k-th iteration point q k The direction of the normalized negative gradient of the function;

[0145] When ξ k >0, update the next iteration point using the iterative format of formula (32). When ξ k <0, update the next iteration point using the iterative format of formula (33).

[0146]

[0147]

[0148] Let k = k + 1, and repeat equations (31)(32)(33) until the 3rd generation. The iteration stops when the convergence condition is met. Taking the first constraint (maximum stress constraint of the stiffened tube) as an example, its iteration history is shown in Table 3;

[0149] Table 3. Iteration history of the ASSA algorithm for solving the maximum stress constraint of stiffened tube structures in the inner layer reliability analysis.

[0150] 1 (0.2482,0.9687) 2.9425 (1.8686,0.8912) 2 (0.2232,0.9747) 2.9417 (1.8748,0.8850) 3 (0.2179,0.9760) 2.9417 (1.8762,0.8836)

[0151] S10. Based on the function value and corresponding gradient information at the performance points of interest, the new outer design points are determined using the sequential quadratic programming method in Matlab.

[0152] S11. Repeat S9-S10. When iterating to the 9th generation, the relative step size between two adjacent design points is 1.4570E-9, satisfying the convergence condition and stopping the iteration. The iteration history of the design points and the quality of the stiffened tube structure is shown in Table 4. Among them, the optimal design results for the equivalent skin thickness and the equivalent stiffener height are d. * = [3.3132, 3.7100] mm, the optimal mass of the reinforced tube structure is 670.58 kg.

[0153] Table 4 Iteration history of design points and objective function

[0154]

[0155]

[0156] To compare accuracy, the MVE method was used to perform uncertainty quantification and reliability optimization design on this embodiment. The data used in MVE is the same as that in the method of this invention, which is the initial data in Table 1. At the same time, the non-probabilistic reliability optimization process is also the same as that in the method of this invention. Figure 6This paper compares the results of ellipsoidal models constructed using two different methods. It can be observed that, under insufficient sample conditions (10 sets of data), the bootstrap-based extended ellipsoidal model proposed in this invention can completely encompass the uncertainty domain of the real samples at a 99% confidence level, while the traditional MVE method cannot even completely encompass the uncertainty domain of the real samples at a 70% confidence level. Therefore, the method proposed in this invention is more accurate than the MVE method.

[0157] Table 5 compares the optimal solutions of the two methods. The baseline solution is the optimal result obtained using the first-order second-moment method under the exact probabilistic model, and its failure probability is consistent with the given target reliability index. (The failure probability is approximately 0.0013), which is basically consistent. It can be observed that the traditional MVE method produces the smallest optimal stiffened tube structure mass, but the failure probability of each constraint is significantly higher than the given value, with the failure probability of the maximum stress constraint exceeding 0.1. Using this method for structural design will result in a significant risk of failure. In contrast, the failure probability of each constraint in the BEEM proposed in this invention is lower than the given value, ensuring that the structure meets design requirements. Although the optimal design's structural mass is slightly higher (4.42%) than the baseline value (642.17 kg), a slightly conservative design is unavoidable considering that only 10 sets of initial data were used in this embodiment. With an increase in the amount of initial data, the BEEM proposed in this invention will achieve even better results.

[0158] Table 5 Comparison of the best results of the BEEM proposed in this invention and the traditional MVE method.

[0159]

[0160]

[0161] Due to the high manufacturing and testing costs of complex aerospace structures, it is difficult to obtain sufficient data in practice to construct accurate uncertainty quantification models, resulting in a significant risk of failure in optimal structural reliability design. To address this issue, this invention introduces the bootstrap resampling concept based on nonprobabilistic modeling theory, proposing a method for uncertainty quantification and structural reliability optimization under insufficient data. This method adaptively establishes an extended ellipsoidal uncertainty domain based on the amount of initial data, achieving high-precision quantification of multi-source uncertainties under insufficient data. Then, combined with the ASSA algorithm, reliable results that meet design requirements are obtained. This invention solves the problem that traditional nonprobabilistic ellipsoidal models cannot accurately quantify uncertainty boundaries under insufficient data conditions, leading to the failure of reliability optimization results. It can provide a lightweight design scheme that meets reliability requirements for the design of aerospace main load-bearing structures under conditions of insufficient actual data.

[0162] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications to the technical solutions described in the foregoing embodiments, or equivalent substitutions for some or all of the technical features, do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for uncertainty quantification and structural reliability optimization for insufficient data, characterized in that, The method includes the following steps: S1. Based on the reliability optimization design requirements of aerospace structures, define a mathematical model for reliability optimization, including determining the objective function, constraint function, design variables, random variables, and setting the target reliability index and initial design point; S2. For the mathematical optimization model obtained in step S1, perform non-probabilistic uncertainty quantification under insufficient data, and collect relevant data of random variables to form an initial dataset. Where ns is the number of initial data points, and the sample mean and covariance of the initial dataset are calculated; eigenvalue analysis is performed on the sample covariance of the initial dataset to obtain the eigenvector matrix, which is the direction of the semi-major axis of the initial ellipsoid; the total number of bootstrap samplings is set to Nsim; and the upper limit of one-sided confidence is set. Set the number of samples, k = 1. S3. Use Bootstrap to perform a resampling to obtain the resampled dataset. Calculate the sample mean and covariance of the resampled dataset, and construct the Bootstrap ellipsoid model based on the gradient optimization algorithm. S4. Approximately uniformly discretize the boundary of the above bootstrap ellipsoid model to obtain a set of boundary points. Construct the boundary point ellipsoid model with the direction of the initial ellipsoid semi-major axis as the given direction. Solve for the optimal ellipsoid semi-major axis length of the boundary point ellipsoid model based on the gradient optimization algorithm. S5. Let kk = kk + 1, and repeat the above steps S3-S4 until kk = Nsim, to obtain the optimal semi-major axis length of the boundary point ellipsoid for each sampling. S6. Statistically determine the optimal ellipsoid semi-major axis length distribution of the boundary point ellipsoid model, eliminate extreme cases using an outlier filtering mechanism, and further obtain the ellipsoid semi-major axis length under a specified confidence level using the upper limit of one-sided confidence and construct an extended ellipsoid model. S7. Based on the extended ellipsoid model, transform the random variables obtained in step S1 to the standard q-space; S8. Based on the extended ellipsoid model, define the structural nonprobabilistic reliability optimization formula, and carry out nonprobabilistic reliability optimization based on the initial design point set in step S1. S9. Based on the current design point, use the first second-order moment algorithm to solve the inner layer reliability analysis in non-probabilistic reliability optimization and obtain the performance points of interest. S10. Based on the function value and gradient information at the performance points of interest, determine the new outer layer design points; S11. Repeat S9-S10 until the convergence condition is met, and obtain the optimal design result with nonprobabilistic reliability.

2. The uncertainty quantification and structural reliability optimization method for insufficient data according to claim 1, characterized in that, In step S1: the mathematical model for structural reliability optimization is as follows: As shown; In the formula, d is the objective function; d is the design variable; X is a random variable; This is the j-th constraint / function; Let be the failure probability of the j-th functional function; m is the number of functional functions; Let be the target reliability index for the j-th functional function; and These are the upper and lower bounds of the design variables; the initial design point is set as... .

3. The uncertainty quantification and structural reliability optimization method for inadequate data according to claim 1, wherein in step S1, the objective function includes, but is not limited to, structural mass or overall structural stiffness; and the constraint function is the failure probability of the structural mechanical response, including the static, dynamic, or buckling response characteristics of the structure.

4. The uncertainty quantification and structural reliability optimization method for insufficient data according to claim 1, characterized in that, In steps S3 and S4, constructing the bootstrap ellipsoid and boundary point ellipsoid requires solving the optimization formula using gradient-based optimization algorithms, including but not limited to the interior point method, the effective set method, or the sequential quadratic programming method; specifically: In step S3: Resampling using Bootstrap involves sampling with replacement from the initial dataset while keeping the data size constant, resulting in the k=1th Bootstrap sample dataset. ; Calculate the k=1th bootstrap sample dataset mean Covariance ; With the length of the semi-major axis of the ellipsoid As the design variable, minimizing the ellipsoidal volume is the objective function. An optimization formula is constructed and solved given the mean. and the direction of the semi-major axis of the ellipsoid The minimum volume ellipsoid of the lower envelope initial data is used to obtain the bootstrap ellipsoid model, as shown in the formula. As shown; In the formula, The bootstrap ellipsoid feature matrix is ​​given by the k=1th sampling; the direction of the semi-major axis of the ellipsoid. It can be derived from the covariance matrix get; This is the inverse of the determinant of the bootstrap ellipsoid characteristic matrix, which, since it is proportional to the volume of the ellipsoid, minimizes... Equivalent to minimizing the volume of the ellipsoid; This represents the i-th initial data; This represents the mean of the k=1th bootstrap sample dataset; The formula is solved using the sequential quadratic programming method in Matlab. The solution is performed to obtain the semi-major axis length of the optimal bootstrap ellipsoid. ; Construct a bootstrap ellipsoid model, whose boundary equations are as follows: As shown; In the formula, the optimal characteristic matrix of the bootstrap ellipsoid under the k=1th sampling is... , as in the formula As shown; In step S4: By boundary equations Approximately uniform sampling is used to obtain a set of boundary points. ;in The number of boundary points; With the length of the semi-major axis of the ellipsoid As the design variable, minimizing the volume of the boundary point ellipsoid is the objective function. An optimization formula is constructed and solved for the given mean. and the direction of the semi-major axis of the ellipsoid The minimum volume ellipsoid of the lower envelope boundary point data, i.e., the boundary point ellipsoid, is obtained as shown in the formula. As shown; In the formula, This is the ellipsoidal feature matrix of the boundary point corresponding to the kk=1th sampling; The direction of the semi-major axis of the initial data ellipsoid; This is the inverse of the determinant of the characteristic matrix of the ellipsoid at the boundary points. Since it is proportional to the volume of the ellipsoid, minimizing it is possible. This is equivalent to minimizing the volume of the boundary point ellipsoid; This represents the i-th boundary point of the optimal boostrap ellipsoid obtained by the kk=1th sampling. The formula is solved using the sequential quadratic programming method in Matlab. By solving the problem, we obtain the optimal semi-major axis length of the boundary point ellipsoid corresponding to the kk=1th sampling. .

5. In the uncertainty quantification and structural reliability optimization method for insufficient data according to claim 1, in step S6, an outlier screening mechanism is used to exclude extreme cases, including the 3σ criterion, box plot method, cluster analysis method, or kernel density estimation method.

6. In the uncertainty quantification and structural reliability optimization method for insufficient data according to claim 2, in step S8, the structural non-probabilistic reliability optimization based on the concern performance method can be defined as the formula... Essentially, it is a nested optimization problem with two layers: the outer layer minimizes the objective function, while the inner layer performs a non-probabilistic reliability analysis; based on the initial design point set in step S1. Conduct nonprobabilistic reliability optimization; In the formula, Corresponding formula The j-th function value in the q-space; Corresponding formula The performance value of the j-th function in the function measure method is equivalent to the performance value of the function corresponding to the minimum performance target point, which can be obtained by solving the formula. Calculated; In the formula, Let be the target nonprobabilistic reliability index for the j-th functional function.

7. The uncertainty quantification and structural reliability optimization method for inadequate data according to claim 1, wherein in step S9, the first-order second-moment algorithm includes, but is not limited to, the enhanced step size adjustment method ASSA.

8. In the uncertainty quantification and structural reliability optimization method for inadequate data according to claim 1, in step S10, a new outer design point is determined by sequential quadratic programming in Matlab based on the function value and corresponding gradient information at the performance point of interest.

9. The uncertainty quantification and structural reliability optimization method for insufficient data according to claim 1, wherein the convergence condition in step S11 is: stopping the iteration until the relative error between two consecutive iteration points does not exceed 1E-6, and obtaining the optimal design result for nonprobabilistic reliability.

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