RBDO method and system based on PC-Kriging model and quantile evaluation method

Abstract

The present invention discloses a RBDO method and system based on a PC-Kriging model and a quantile evaluation method, wherein the method includes the following steps S1-S2: Step S1, using a reliability analysis algorithm based on an improved adaptive PC-Kriging model to calculate the reliability of a target object; Step S2, based on step S1, performing RBDO optimization calculation to obtain an optimization result; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, using a combined point addition strategy of the MP criterion combined with the EI criterion to make the proxy model approach the objective function to the optimal solution, and introducing a local refinement strategy based on a rough probability constraint proxy model established by a global enrichment strategy so that the probability constraint proxy model can accurately evaluate the quantile of each iterative solution during the optimization process. The method adopted by the present invention can greatly reduce the number of calls to the original model while ensuring accuracy, greatly improve the calculation speed, and has strong adaptability.

Description

Technical Field

[0001] The present invention relates to the technical field of reliability optimization calculation of a target object, and in particular to a RBDO method and system based on a PC-Kriging model and a quantile evaluation method. Background Art

[0002] RBDO technology has been widely studied and applied in recent years. It seeks a good balance between optimization objectives and safety requirements by moving the solution out of the constraint boundary and searching inside the admissible space.

[0003] For the RBDO problem, the conventional solution is to solve the two-layer loop nesting problem, and its computational cost is prohibitive. Therefore, domestic and foreign scholars have conducted extensive research on the solution of the RBDO problem and formed three mainstream solution strategies, namely the RIA method and the PMA method based on approximation technology, and the SORA method based on decoupling technology. However, all three methods rely on repeated evaluation of the original calculation model, which restricts their application in complex engineering problems, and they all expect reliability and optimization problems to converge to the optimal goal at the same time, which is bound to bring about problems of algorithm stability, especially when the initial design is far away from the optimal design point and the limit state function is highly nonlinear, this situation will be more serious.

[0004] Proxy model technology can greatly reduce the computational burden of reliability optimization. Therefore, the application of surrogate models in RBDO has been widely studied. They are all committed to improving the efficiency of surrogate models in solving reliability optimization problems, so that they can quickly converge to the accurate optimal solution with as few training samples as possible and obtain a high-precision approximation of the original model in the optimal solution sub-region.

[0005] In the prior art, the RBDO method combining the proxy model and MCS mostly uses the failure indicator function to evaluate the failure probability, which will bring instability to the calculation of the RBDO model; first, the sensitivity calculation of the probability constraint to each random variable involves all MCS samples, and these sensitivities are predicted by the proxy model, so a large number of training samples are required to ensure the high accuracy of the sensitivity prediction. Secondly, the discontinuity of the failure indicator function in the MCS needs to be improved by an additional smoothing approximation strategy, and when the failure probability constraint is sufficiently satisfied or violated, the sensitivity of the failure probability constraint to the corresponding variable is close to zero, causing the optimizer to stagnate.

[0006] In the prior art, patent CN114662370 provides an adaptive PCK method for time-varying structural reliability analysis. The patent mainly involves building a PCK model of a turbine engine and performing reliability analysis based on it. The patent does not involve the RBDO problem and does not provide corresponding inspiration. Summary of the invention

[0007] Purpose of the invention: In order to overcome the deficiencies in the prior art, the present invention provides an RBDO method and system based on the PC-Kriging model and quantile evaluation method, which can greatly reduce the number of calls to the original model and has strong adaptability and is applied to integrated mechanical devices.

[0008] Technical solution: To achieve the above-mentioned purpose, the RBDO method based on the PC-Kriging model and the quantile evaluation method of the present invention comprises the following steps S1-S2:

[0009] Step S1, calculating the reliability of the target object using a reliability analysis algorithm based on an improved adaptive PC-Kriging model;

[0010] Step S2, based on the calculated reliability of the target object, RBDO optimization calculation is performed to obtain an optimization result; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, adopting a combined point adding strategy of the MP criterion and the EI criterion to make the proxy model approach the objective function to the optimal solution, and introducing a local refinement strategy on the basis of a rough probability constraint proxy model established by a global enrichment strategy so that the probability constraint proxy model can accurately evaluate the quantile of each iterative solution during the optimization process.

[0011] The step S1 includes the following steps S101-S106:

[0012] Step S101, based on the random distribution of input parameters, generate N MCS Initial candidate design samples, and these sample sets are recorded as

[0013] Step S102, from A small number of sample points N0 are randomly selected as the initial training samples for constructing the PC-Kriging model, and the training points are expressed as And calculate the corresponding The true response value Get the training sample set

[0014] Step S103: based on the training sample set Construct a PC-Kriging model; and perform a PC-Kriging model on the sample set based on the obtained PC-Kriging model. Analyze each sample point x in and get the corresponding response and variance And calculate the estimated failure probability based on the original calculation model Wherein, the PC-Kriging model is:

[0015]

[0016] Among them, y=Y(x) is a specific implementation of a Gaussian static random process, and its domain is Y (PCK) (x) represents the PC-Kriging model; is the weighted sum of orthogonal polynomials describing the trend of the PC-Kriging model, α=[α1 α2…α M ] T is composed of M non-negative integers α i The vector composed of is the set of all possible α, which is determined by the truncation strategy, y α are the coefficients of the polynomial expansion, is a multivariate polynomial orthogonal to the probability density function corresponding to the input parameters, is the definition of input parameters in chaotic polynomial space; σ 2 is the variance of the Gaussian process, Z(x, ω) is a stationary Gaussian process with zero mean and unit variance, and ω represents the potential probability space;

[0017] Step S104: using the adaptive point selection strategy, The best training sample x* is selected; the adaptive point selection strategy is based on the following learning function:

[0018]

[0019] Where p(x) is the joint probability density of random variables at point x; U(x) is the reliability index associated with the probability of misclassification; w pro (x) is the probability weight function; U mod (x) is the revised reliability index:

[0020] The selection criteria for the best test points are:

[0021]

[0022] Step S105, determine whether the current iteration meets the stopping criteria, if so, go to step S106; otherwise, calculate the best training sample x * The corresponding true response value y * , x * and * Add to Sample Set middle, Then return to step S103;

[0023] Step S106: Use the test formula to test the sample Is the total number of failures sufficient to accurately estimate the failure probability? If the test formula is satisfied, the final failure probability is output. And get the reliability of the target object; otherwise, it means N MCS The number of candidate design samples is not enough. Increase ΔN MCS candidate design samples, and repeat step S103; the result of the formula is:

[0024]

[0025] in, The failure probability The corresponding coefficient of variation, COV thr is the coefficient of variation threshold.

[0026] Furthermore, the step S2 includes the following steps S201-S210:

[0027] Step S201, input the initial design point D (0) , set the maximum number of optimization iterations it max , generating the augmented reliability space

[0028] Step S202, constructing an initial proxy model; using uniform distribution to extract (2n d +1) LHS points, n d is the dimension of the input variable of the corresponding function; these sampling points are used to generate the corresponding proxy model, and the proxy model of the objective function is defined as The corresponding test sample set is recorded as Surrogate Models for Probabilistic Constraint Functions The corresponding test sample set is recorded as Surrogate Models for Deterministic Constraint Functions The corresponding test sample set is recorded as

[0029] Step S203, running the global proxy model enrichment strategy to generate a relatively rough proxy model of the probability constraint function before optimization; adaptively adding sample points with the minimum number of deviations to the corresponding test sample set according to the global proxy model enrichment strategy until the relaxed convergence criterion described by the formula is met;

[0030] Step S204, initialize the optimization program, set the optimization iteration number s=1, and set the initial training indicator function for all proxy models of the probability constraint function The proxy model i representing the corresponding performance function needs to be refined locally, and When , there is no need to refine the local proxy model;

[0031] Step S205, running the optimization program to calculate the RBDO optimization problem, based on the MP criterion combined with the EI criterion combined point adding strategy and the calculation model of the RBDO optimization problem to obtain the sub-optimization problem, based on the current and Solve the sub-optimization problem to approximately obtain the optimal solution or optimal solution set D of the objective function (s) , and set s=s+1;

[0032] Step S206, determine whether one of the following two conditions is met, if yes, proceed to step S210, otherwise proceed to step S207; wherein condition 1 is: || D (s) -D (s-1) ||≤1.0×10 -3 and Condition 2: s ≥ it max Among them, D (s) With D (s-1) are two adjacent optimal solutions. and is the value of the qth objective function in the s-1th round and the sth round;

[0033] Step S207, update the proxy model of the objective function and the deterministic constraint function; for the proxy model of the objective function, if the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than Then add the optimal design point to the training sample set, otherwise, skip the optimal design point, where is the reliability index of the i-th probability constraint; the proxy model for the deterministic constraint function j If the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than Then add the optimal design point as a training point, otherwise, skip the optimal design point;

[0034] Step S208: Update the training instruction function When the current optimal solution D is determined (s) The optimal solution D of the previous iteration (s-1) The distance in standard normal space is greater than Then set the corresponding The value is 1;

[0035] Step S209, updating the probability constraint function proxy model; The corresponding proxy model runs the local proxy model refinement strategy, updates the corresponding constraint proxy model, sets s=s+1, and returns to step S205;

[0036] Step S210, end the program and output the final optimization result.

[0037] Furthermore, the step S203 specifically includes the following steps:

[0038] Step S301: uniformly sample a set of candidate design variable samples Where D is a set of design variables including randomness and determinism;

[0039] Step S302: for each candidate design variable sample Follow these steps:

[0040] a) Generate N number of s The corresponding sample set

[0041] Among them, x (k) is a random design variable, z (k) is an environmental random variable;

[0042] b) Calculate the sample set The proxy model response

[0043] c) Calculate the proxy model response set The quantile q α (D (j) );

[0044] d) Determine the sample points associated with the quantiles

[0045] in, For PC-Kriging model The average value of

[0046] e) Calculate the sample points associated with the quantiles U-function:

[0047]

[0048] Among them, u(D (j) ) is a learning function related to quantiles; For U-function, is the failure threshold; is the standard deviation;

[0049] Step S303: Select all candidate design variable samples The point with the smallest deviation is added as the best improved design point to the experimental sample points of the PC-Kriging model:

[0050]

[0051] Step S303, when the number of changes in the sign of the quantile predicted by the PC-Kriging model is small enough in two consecutive enrichment iterations, the algorithm operation is stopped, and it is considered that the PC-Kriging model meets the accuracy requirements at this time; the relaxed convergence criterion here is defined as:

[0052]

[0053] Among them, ε sc is the convergence threshold.

[0054] Furthermore, the step S204 includes the following steps:

[0055] Step S401, using the Monte Carlo sampling method to generate the design point D corresponding to the current optimization iteration number s (s) Sample set

[0056] Step S402: Based on the current proxy model Calculate the quantile of the current design point and its corresponding upper and lower boundaries and and Corresponding to and quantile of ;

[0057] Step S403, determining the training instruction function IND i , i=1,2…,n h Is it equal to 1? i =1, the corresponding agent model i performs subsequent operations accordingly;

[0058] Step S404, skip the training of the proxy model with non-active constraints; set the number of iterations, and when s is greater than the number of iterations, the training indicator function IND of the non-active constraints is set. i Set to 0;

[0059] Step S405, add training sample sets and refine the accuracy of the local PC-Kriging proxy model until the local accuracy criterion of the proxy model is met, then IND i Set to 0; the local accuracy criterion of the proxy model is:

[0060]

[0061] In the formula, is the preset accuracy threshold; when the failure threshold When Substituting the denominator, where is the PC-Kriging proxy model in the augmented space sample set The variance of

[0062] Step S406: When all IND i are all set to 0, and the optimization procedure in step S205 is executed.

[0063] The step S405 specifically includes:

[0064] Step S501, referring to the learning function in step S104 to calculate {u k (D (s) ),1≤k≤N s}; Specifically:

[0065]

[0066] Among them, u k (D (s) ) is the learning function corresponding to the kth sample; p(·) is the joint probability density of random variables, where the deviation number U can be calculated by the following formula:

[0067]

[0068] Step S502: Select the sample set of the current design point in, u k (D (s) ) value is the best improved design point and is added to the experimental sample points of the PC-Kriging model:

[0069]

[0070] The present invention also provides a RBDO system based on the PC-Kriging model and the quantile evaluation method, the system comprising:

[0071] A reliability calculation module, which is used to calculate the reliability of the target object using a reliability analysis algorithm based on an improved adaptive PC-Kriging model;

[0072] An optimization calculation module is used to perform RBDO optimization calculation to obtain an optimization result based on the calculated reliability of the target object; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, adopting a combined point adding strategy of the MP criterion and the EI criterion to make the proxy model approach the optimal solution of the objective function, and introducing a local refinement strategy based on the rough probability constraint proxy model established by the global enrichment strategy to enable the probability constraint proxy model to accurately evaluate the quantile of each iterative solution during the optimization process.

[0073] Beneficial effects: The RBDO method and system based on the PC-Kriging model and quantile evaluation method of the present invention have the following beneficial effects:

[0074] (1) The adopted method can greatly reduce the number of calls to the original model while ensuring accuracy, reduce the occupation of computing resources and the number of iterations of the optimization process, and greatly improve the calculation speed. It has strong adaptability to the reliability optimization problem of the target object.

[0075] (2) The PC-Kriging model is introduced into the reliability calculation, and the joint probability density of random variables is introduced as the weight function and integrated into the learning function. Sample points with a higher probability of misclassification are given a larger weight, which has a greater impact on the reliability calculation and can greatly improve the convergence speed of the proxy model used for reliability analysis.

[0076] (3) By setting up an augmented space for the RBDO problem to be solved, a single global proxy model of the corresponding function can be constructed by designing appropriate experimental sample points in the augmented space for the full solution of subsequent RBDO problems, without the need to repeatedly construct proxy models during the optimization process.

[0077] (4) By introducing the concept of quantile, the measure of probability constraints in the traditional RBDO model is equivalently converted into the form of quantile, and then the feasibility of the probability constraints is judged according to the sign of the quantile. This operation can effectively avoid the drawbacks of using indicator functions to calculate probability constraints.

[0078] (5) The traditional “MP+EI” combined point strategy has strong comprehensive performance. However, when processing close sample points, this combined strategy may cause the correlation matrix to be “ill-conditioned” and reduce the model prediction accuracy. To address the above problem, the present invention introduces the idea of ​​“skipping scheme” to effectively avoid the above phenomenon. Specifically, by optimizing the single objective function to obtain the new sample points, arrive The distance in standard normal space is less than Then only one of the two samples is retained and added to the training sample set. Similarly, for the new sample set obtained by multi-objective function optimization and If there are sample points whose distance in the standard normal space is less than Only one of the two sample points is retained. The above method effectively avoids the reduction of model prediction accuracy caused by sample points that are too close.

[0079] (6) The global surrogate model enrichment strategy before optimization can find the design points that are most likely to improve the performance of the surrogate model to gradually obtain a relatively accurate surrogate model, thereby avoiding the crude use of a large number of space-filling experimental sample points to obtain accurate model information.

[0080] (7) Through the local refinement strategy of the probability constraint surrogate model in the optimization process, the optimization is combined with the surrogate model enrichment, which solves the remaining epistemic uncertainty problems introduced by the PC-Kriging surrogate model and ensures the accuracy of the evaluation of the quantile of each iterative solution in the optimization process. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 A schematic diagram of the process of calculating the reliability of a target object using a reliability analysis algorithm based on an improved adaptive PC-Kriging model;

[0082] Figure 2 A schematic diagram of the process of performing RBDO optimization calculation to obtain the optimization result;

[0083] Figure 3 This is the result diagram of Example 1;

[0084] Figure 4 This is a schematic diagram of the gearbox configuration in Example 2. DETAILED DESCRIPTION

[0085] The present invention will be further described below in conjunction with the accompanying drawings.

[0086] The RBDO method based on the PC-Kriging model and the quantile evaluation method of the present invention comprises the following steps S1-S2:

[0087] Step S1, using a reliability analysis algorithm based on an improved adaptive PC-Kriging model to calculate the reliability of a target object; wherein the target object is an integrated mechanical device;

[0088] Step S2, based on the calculated reliability of the target object, RBDO optimization calculation is performed to obtain an optimization result; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, adopting a combined point adding strategy of the MP criterion and the EI criterion to make the proxy model approach the objective function to the optimal solution, and introducing a local refinement strategy on the basis of a rough probability constraint proxy model established by a global enrichment strategy so that the probability constraint proxy model can accurately evaluate the quantile of each iterative solution during the optimization process.

[0089] Specifically, Figure 1 As shown, the step S1 includes the following steps S101-S106:

[0090] Step S101, based on the random distribution of input parameters, generate N MCS Initial candidate design samples, and these sample sets are recorded as

[0091] Step S102, from A small number of sample points N0 are randomly selected as the initial training samples for constructing the PC-Kriging model, and the training points are expressed as And calculate the corresponding The true response value Get the training sample set

[0092] Step S103: based on the training sample set Construct a PC-Kriging model; and perform a PC-Kriging model on the sample set based on the obtained PC-Kriging model. Analyze each sample point x in and get the corresponding response and variance And calculate the estimated failure probability based on the original calculation model Wherein, the PC-Kriging model is:

[0093]

[0094] Among them, y=Y(x) is a specific implementation of a Gaussian static random process, and its domain is Y (PCK) (x) represents the PC-Kriging model; is the weighted sum of orthogonal polynomials describing the trend of the PC-Kriging model, α=[α1α2…α M ] T is composed of M non-negative integers α i The vector composed of is the set of all possible α, which is determined by the truncation strategy, y α are the coefficients of the polynomial expansion, is a multivariate polynomial orthogonal to the probability density function corresponding to the input parameters, is the definition of input parameters in chaotic polynomial space; σ 2 is the variance of the Gaussian process, Z(x,ω) is a stationary Gaussian process with zero mean and unit variance, and ω represents the potential probability space;

[0095] Step S104: using the adaptive point selection strategy, Filter out the best training sample x * ; The adaptive point selection strategy is based on the following learning function:

[0096]

[0097] Where p(x) is the joint probability density of random variables at point x; U(x) is the reliability index associated with the probability of misclassification; w pro (x) is the probability weight function; U mod (x) is the revised reliability index;

[0098] The selection criteria for the best test points are:

[0099]

[0100] Step S105, determine whether the current iteration meets the stopping criteria, if so, go to step S106; otherwise, calculate the best training sample x * The corresponding true response value y * , x * and * Add to Sample Set middle, Then return to step S103;

[0101] Step S106: Use the test formula to test the sample Is the total number of failures sufficient to accurately estimate the failure probability? If the test formula is satisfied, the final failure probability is output. And get the reliability of the target object; otherwise, it means N MCS The number of candidate design samples is not enough. Increase ΔN MCS candidate design samples, and repeat step S103; the result of the formula is:

[0102]

[0103] in, The failure probability The corresponding coefficient of variation, COV thr is the coefficient of variation threshold.

[0104] In the above steps S101-S106, the PC-Kriging model is introduced into the reliability calculation, and by introducing the joint probability density of random variables as a weight function, it is integrated into the learning function, and those sample points with a higher probability of misclassification are given a larger weight. They have a greater impact on the reliability calculation and can greatly improve and accelerate the convergence speed of the proxy model used for reliability analysis.

[0105] like Figure 2 As shown, the step S2 includes the following steps S201-S210:

[0106] Step S201, input the initial design point D (0) , set the maximum number of optimization iterations it max , generating the augmented reliability space

[0107] In this step, since the optimization process design variable D = (X, d d )(including random design variables X and deterministic design variables d d ) is constantly changing. Rebuilding the proxy model in each iteration process will increase the computational burden. Therefore, using a single proxy model in the design area will make the problem simpler and greatly improve the solution efficiency. In this step, the augmented reliability space technology is used to construct the proxy model of the target device; the augmented space can be defined as two hyperrectangular confidence regions The tensor product between the design variables D = (X, d d ) It can be obtained by the following tensor product:

[0108]

[0109] Where M d is the dimension of the random design variable, and are the quantiles associated with the lower and upper limits of the design variable, respectively, and Designing for randomness i At the extreme lower limit of the design space and upper limit The inverse cumulative distribution function at α i is the probability that the random design variable is sampled outside the augmented space of the i-th dimension.

[0110] Confidence region associated with the environmental random variable Z is defined as:

[0111]

[0112] Where M z is the dimension of the environmental random variable, and are the quantiles associated with the lower and upper limits of the design variable, α j is the probability that the environmental random variable is sampled outside the j-th dimension of the augmented space.

[0113] Step S202, constructing an initial proxy model; using uniform distribution to extract (2n d +1) LHS points, ensuring that the PC-Kriging model will be more robust when there are at least 12 LHS points at the beginning. d is the dimension of the input variable of the corresponding function; it should be noted that the design domain of the probability constraint function is expanded to the augmented reliability space; these sampling points are used to generate the corresponding surrogate model, and the surrogate model of the objective function is defined as The corresponding test sample set is recorded as Surrogate Models for Probabilistic Constraint Functions The corresponding test sample set is recorded as Surrogate Models for Deterministic Constraint Functions The corresponding test sample set is recorded as

[0114] Step S203, running the global proxy model enrichment strategy to generate a relatively rough proxy model of the probability constraint function before optimization; adaptively adding sample points with the minimum number of deviations to the corresponding test sample set according to the global proxy model enrichment strategy until the relaxed convergence criterion described by the formula is met;

[0115] Step S204, initialize the optimization program, set the optimization iteration number s=1, and set the initial training indicator function for all proxy models of the probability constraint function The proxy model i representing the corresponding performance function needs to be refined locally, and When , there is no need to refine the local proxy model;

[0116] Step S205, running the optimization program to calculate the RBDO optimization problem, based on the MP criterion combined with the EI criterion combined point adding strategy and the calculation model of the RBDO optimization problem to obtain the sub-optimization problem, based on the current and Solve the sub-optimization problem to approximately obtain the optimal solution or optimal solution set D of the objective function(s) , and set s=s+1;

[0117] In this step, the calculation model of the RBDO optimization problem is:

[0118] Find:{μ X ,d d}

[0119] min:f(μ Z ,μ X ,d d )={f1(μ Z ,μ X ,d d ),...,f q (μ Z ,μ X ,d d ),...,f K (μ Z ,μ X ,d d )}

[0120]

[0121] and / or:g j (d d )≤0,j=1,2,...,n h

[0122]

[0123] Among them, f is the objective function vector, d d is the vector of deterministic design variables, Z is the environmental random variable, X(d) is the vector of random design variables, d is the given design variable, μ X , μ Z Represent the average values ​​of X and Z respectively; P r (·) is the probability function, PF i T and β i T Respectively represent the satisfaction of the i-th probability constraint (hard constraint) is the allowable failure probability and reliability index; Φ(·) is the standard normal cumulative distribution function, is the jth deterministic constraint (soft constraint), and the superscripts L and U represent the lower and upper limits of the design variables, respectively;

[0124] The MP criterion is to find the minimum value of the objective function directly on the proxy model. When this criterion is selected, the constructed proxy model is directly used. To replace the objective function f(μZ ,μ X ,d d ), that is, solve the following sub-optimization problem;

[0125] Find:

[0126] min:

[0127] Subject to:

[0128] and / or:

[0129] where

[0130] The superscript MP represents that what is sought is the optimal solution of the MP criterion.

[0131] The EI criterion combines the mean and error information provided by the proxy model at the prediction point to predict the optimal point in the search domain. Assume that the current qth objective function f q The minimum value of It is considered that the objective function response value predicted by the PC-Kriging surrogate model is a subject to the mean The standard deviation is s q The probability density of a normally distributed random variable (x) can be expressed as:

[0132]

[0133] For the problem of minimizing the objective value, the improvement I q (x) is defined as:

[0134]

[0135] I q (x) The expected value can be calculated by the following formula:

[0136]

[0137] Where Φ and φ are the standard normal cumulative distribution function and the standard normal distribution probability density function, respectively.

[0138] Then we can get the expected expression of improvement when there are multiple objective functions:

[0139] E[I(x)]={E[I1(x)],...,E[I q (x)],...,E[I K (x)]};

[0140] Among them, E[I(x)] is the expected improvement of the objective function, I q is the improvement value of the qth objective function.

[0141] When the EI criterion is selected, it is necessary to replace Min:f(μ Z , μ X , d d ) is replaced by Max:E[I(x)], that is, solving the following sub-optimization problem:

[0142] Find:

[0143] m1n:

[0144] Subject to:

[0145] and / 0r:

[0146] where

[0147] In the formula, the superscript EI represents that what is sought is the optimal solution of the EI criterion.

[0148] In this step, when performing optimization calculations, a combined optimization algorithm of genetic algorithm and sequential quadratic programming is used to solve single-objective optimization problems, while a non-dominated sorting genetic algorithm is used to solve multi-objective optimization problems.

[0149] Step S206, determine whether one of the following two conditions is met, if yes, proceed to step S210, otherwise proceed to step S207; wherein condition 1 is: || D (s) -D (s-1) ||≤1.0×10 -3 and Condition 2: s ≥ it max Among them, D (s) With D (s-1) are two adjacent optimal solutions. and is the value of the qth objective function in the s-1th round and the sth round;

[0150] Step S207, update the proxy model of the objective function and the deterministic constraint function; for the proxy model of the objective function, if the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than Then add the optimal design point to the training sample set, otherwise, skip the optimal design point, where is the reliability index of the i-th probability constraint; the proxy model for the deterministic constraint function j If the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than Then add the optimal design point as a training point, otherwise, skip the optimal design point;

[0151] Step S208: Update the training instruction function When the current optimal solution D is determined (s) The optimal solution D of the previous iteration (s-1) The distance in standard normal space is greater than Then set the corresponding The value is 1;

[0152] Step S209, updating the probability constraint function proxy model; The corresponding proxy model runs the local proxy model refinement strategy, updates the corresponding constraint proxy model, sets s=s+1, and returns to step S205;

[0153] Step S210, end the program and output the final optimization result.

[0154] In the above steps, an augmented space is set for the RBDO problem to be solved. By designing appropriate experimental sample points in the augmented space, a single global proxy model of the corresponding function can be constructed for the entire solution of subsequent RBDO problems, without the need to repeatedly construct the proxy model during the optimization process.

[0155] By introducing the concept of quantile, the measure of probability constraints in the traditional RBDO model is equivalently converted into the form of quantile, and then the feasibility of the probability constraint is judged according to the sign of the quantile. This operation can effectively avoid the drawbacks of using indicator functions to calculate probability constraints.

[0156] The traditional "MP+EI" combined point strategy has a strong comprehensive performance, but this combined strategy may cause the correlation matrix to be "ill-conditioned" when processing close sample points, thereby reducing the model prediction accuracy. The present invention introduces the idea of ​​"skipping scheme" to effectively avoid the above phenomenon. Specifically, by optimizing the single objective function to obtain the new sample arrive The distance in standard normal space is less than Then only one of the two samples is retained and added to the training sample set. Similarly, for the new sample set obtained by multi-objective function optimization and If there are sample points whose distance in the standard normal space is less than Only one of the two sample points is retained. The above method effectively avoids the reduction of model prediction accuracy caused by sample points that are too close.

[0157] Furthermore, the step S203 specifically includes the following steps:

[0158] Step S301: uniformly sample a set of candidate design variable samples Where D is a set of design variables including randomness and determinism;

[0159] Step S302: for each candidate design variable sample Follow these steps:

[0160] a) Generate N number of s The corresponding sample set

[0161] Among them, x (k) is a random design variable, z (k) is an environmental random variable;

[0162] b) Calculate the sample set The proxy model response

[0163] c) Calculate the proxy model response set The quantile q α (D (j) );

[0164] d) Determine the sample points associated with the quantiles

[0165] in, For PC-Kriging model The average value of

[0166] e) Calculate the sample points associated with the quantiles U-function:

[0167]

[0168] Among them, u(D (j) ) is a learning function related to quantiles; For U-function, is the failure threshold; is the standard deviation;

[0169] Step S303: Select all candidate design variable samples The point with the smallest deviation is added as the best improved design point to the experimental sample points of the PC-Kriging model:

[0170]

[0171] Step S303, when the number of changes in the sign of the quantile predicted by the PC-Kriging model is small enough in two consecutive enrichment iterations, the algorithm operation is stopped, and it is considered that the PC-Kriging model meets the accuracy requirements at this time; the relaxed convergence criterion here is defined as:

[0172]

[0173] Among them, ε sc is the convergence threshold. The main purpose of the proxy model constructed above is to locate the limit state surface without excessively pursuing its accuracy index, so the convergence threshold ε sc From less than 10 -3 The level of is relaxed to 0.01-0.02 to improve the convergence speed of the model.

[0174] The global surrogate model enrichment strategy before optimization can find the design points that are most likely to improve the performance of the surrogate model to gradually obtain a relatively accurate surrogate model, thereby avoiding the crude use of a large number of space-filling test sample points to obtain accurate model information.

[0175] The step S204 includes the following steps:

[0176] Step S401, using the Monte Carlo sampling method to generate the design point D corresponding to the current optimization iteration number s (s) Sample set

[0177] Step S402: Based on the current proxy model Calculate the quantile of the current design point and its corresponding upper and lower boundaries and and Corresponding to and quantile of ;

[0178] Step S403, determining the training instruction function IND i ,i=1,2...,n h Is it equal to 1? i =1, the corresponding agent model i performs subsequent operations accordingly;

[0179] Step S404, skip the training of the proxy model of the non-active constraints; in the initial iteration process, the determination of the non-active constraints may not be accurate. To ensure its accuracy, set the number of iterations (10 times is used in this paper). When s is greater than the number of iterations, the training indicator function IND of the non-active constraints isi Set to 0;

[0180] Step S405, add training sample sets and refine the accuracy of the local PC-Kriging proxy model until the local accuracy criterion of the proxy model is met, then IND i Set to 0; the local accuracy criterion of the proxy model is:

[0181]

[0182] In the formula, is the preset accuracy threshold; when the failure threshold When Substituting the denominator, where is the PC-Kriging proxy model in the augmented space sample set The variance of

[0183] Step S406: When all IND i are all set to 0, and the optimization procedure in step S205 is executed.

[0184] Through the local refinement strategy of the probability constraint surrogate model in the above optimization process, the optimization is combined with the surrogate model enrichment, which solves the remaining epistemic uncertainty problems introduced by the PC-Kriging surrogate model and ensures the accuracy of the evaluation of the quantile of each iterative solution in the optimization process.

[0185] The step S405 specifically includes:

[0186] Step S501, referring to the learning function in step S104 to calculate {u k (D (s )),1≤k≤N s}; Specifically:

[0187]

[0188] Among them, u k (D (s )) is the learning function corresponding to the kth sample; p(·) is the joint probability density of random variables, where the deviation number U can be calculated by the following formula:

[0189]

[0190] Step S502: Select the sample set of the current design point in, u k (D (s) ) value is the best improved design point and is added to the experimental sample points of the PC-Kriging model:

[0191]

[0192] The present invention also provides an RBDO system based on the PC-Kriging model and the quantile evaluation method. The system may include or be divided into one or more program modules. One or more program modules are stored in a storage medium and executed by one or more processors to complete the present invention and implement the above-mentioned RBDO method based on the PC-Kriging model and the quantile evaluation method. The program module referred to in the embodiment of the present invention refers to a series of computer program instruction segments that can perform specific functions, which is more suitable for describing the execution process of the RBDO method based on the PC-Kriging model and the quantile evaluation method in the storage medium than the program itself. The following description will specifically introduce the functions of each program module of this embodiment. The system includes:

[0193] A reliability calculation module, which is used to calculate the reliability of a target object using a reliability analysis algorithm based on an improved adaptive PC-Kriging model; wherein the target object is an integrated mechanical device;

[0194] An optimization calculation module is used to perform RBDO optimization calculation to obtain an optimization result based on the calculated reliability of the target object; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, adopting a combined point adding strategy of the MP criterion and the EI criterion to make the proxy model approach the optimal solution of the objective function, and introducing a local refinement strategy based on the rough probability constraint proxy model established by the global enrichment strategy to enable the probability constraint proxy model to accurately evaluate the quantile of each iterative solution during the optimization process.

[0195] In order to verify the effectiveness of the RBDO method based on the PC-Kriging model and quantile evaluation method in the present invention, the performance of the proposed algorithm is verified by two examples below. They are a 2D highly nonlinear RBDO problem, a gear transmission design problem, and a structural welding design problem. The optimization problem of two design objective functions is considered for the structural welding design problem. These examples verify the adaptability of the proposed method in dealing with highly nonlinear, multiple failure regions, multiple failure modes and multi-objective optimization problems.

[0196] Example 1: A 2D highly nonlinear RBDO problem with multiple most likely failure points.

[0197] This problem contains two random design variables and two constraints, one of which is a highly nonlinear function with multiple MPPs, and the other is a linear constraint function. The initial design point is taken as the midpoint of the design variable. Its specific expression is as follows:

[0198] Find:

[0199] min:

[0200] Subject to:

[0201] where

[0202] G1(X)=x1sin(4x1)+1.1sin(2x2)

[0203] G2(X)=3-x1-x2

[0204] In the formula, x m are mutually independent random input variables that follow a standard normal distribution, and define g ex1 Failure occurs when (x)≤0.

[0205] The following table lists the optimization results and computational costs of solving the 2D highly nonlinear problem of Example 1 using different reliability optimization methods. It can be found that the reliability optimization algorithm using the proxy model significantly reduces the number of calls to the original function compared to the SORA and PMA methods that directly use the original function for optimization, and can improve the accuracy of the solution to a certain extent.

[0206]

[0207] The reliability optimization algorithm proposed in the present invention is used to solve the objective function value convergence curve of Example 1 and the experimental design of the constraint function G1(X) and its proxy model approximation, such as Figure 3 As shown in (a), it can be found that when the convergence condition is reached when the objective function is called 18 times, the optimization program is terminated. The optimal solution obtained at this time is extremely close to the Monte Carlo reference optimal solution calculated using 107 samples. Comparing the PC-Kriging proxy model and the real constraint function of the G1(X) constraint function obtained at the end of the algorithm, it is found that a large number of experimental design points are encrypted in the area near the optimal solution, which ensures the accuracy of the proxy model in this local area, while other areas only need to meet the rough approximation of the model. In this example, G2(X) is a non-active constraint, that is, it always maintains a high reliability index during the optimization process, such as Figure 3 As shown in (b), G1(X) is an active constraint containing multiple MPPs. The optimal solution is often in its boundary area. When the optimal design point is obtained, the quantile calculated by the probability constraint of G1(X) is 5.09×10-5, which meets the constraint requirements.

[0208] Example 2: Gear transmission design problem.

[0209] The purpose of gear transmission optimization design is to make the weight of the reducer the lightest under 11 related constraints (gears meet bending and contact stress indicators, support shafts meet stress and longitudinal displacement indicators, and related dimensional constraints). Figure 4 The figure shows the schematic diagram of the gearbox configuration. The seven random design variables are the gear width, the tooth module, the number of pinion teeth, the distance between the two shaft bearings and their diameters. These design variables follow a mutually independent normal distribution, and the initial design points are [3.5, 0.7, 17, 7.3, 7.72, 3.35, 5.29] T The problem can be expressed as follows:

[0210] Find:

[0211] min:

[0212]

[0213] Subject to:

[0214] where

[0215]

[0216]

[0217]

[0218]

[0219] The following table lists the optimization results and computational cost comparison of Example 2 solved by different reliability optimization methods. It can be found that except for the SORA method, which has some differences in the optimal solution, the objective function values ​​and design variable values ​​solved by other methods are very close. Among them, the reliability optimization algorithm using the proxy model significantly reduces the number of calls to the original function compared with the SORA and ASORA methods that directly use the original function for optimization. When the method of the present invention is used to solve the problem, the number of solutions required for the objective function and constraint function is less than that of the SSRBO method. This fully demonstrates that the strategy of ignoring non-active constraints and skipping sample points with too close spatial distances in the reliability optimization algorithm proposed in this paper can effectively reduce the calls to the original function within the accuracy range. At the same time, it should be pointed out that although the SSRBO method is also effective in this problem, the reliability index of the sixth probability constraint at its optimal solution, β6=2.946, does not meet the target reliability requirements, while the quantiles of the four key constraints at the optimal solution of the method proposed in this paper are q5=1.62×10 -5 ,q6=4.19×10-5 ,q8=5.46×10 -5 and q 11 =9.04×10 -6 All meet the constraint index requirements.

[0220] method Optimization results <![CDATA[N obj ]]> N SORA 3040.02(3.580,0.7,17,7.3,7.764,3.366,5.301) 21 500 ASORA 3038.61(3.577,0.7,17,7.3,7.754,3.365,5.301) 22 396 SSRBO 3038.53(3.576,0.7,17,7.3,7.754,3.366,5.301) 39 39 Method of the present invention 3038.72(3.577,0.7,17,7.3,7.754,3.365,5.302) 25 26

[0221] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. The RBDO method based on the PC-Kriging model and the quantile evaluation method is applied to an integrated mechanical device, characterized in that: The method comprises the following steps S1-S2: Step S1, calculating the reliability of the target object using a reliability analysis algorithm based on an improved adaptive PC-Kriging model; Step S2, based on the calculated reliability of the target object, RBDO optimization calculation is performed to obtain an optimization result; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, adopting a combined point adding strategy of MP criterion and EI criterion to make the proxy model approach the optimal solution of the objective function, and introducing a local refinement strategy based on the rough probability constraint proxy model established by the global enrichment strategy to enable the probability constraint proxy model to accurately evaluate the quantile of each iterative solution during the optimization process; The step S1 includes the following steps S101-S106: Step S101, based on the random distribution of input parameters, generate N MCS Initial candidate design samples, and these sample sets are recorded as Step S102, from A small number of sample points N0 are randomly selected as the initial training samples for constructing the PC-Kriging model, and the training points are expressed as And calculate the corresponding The true response value Get the training sample set Step S103: based on the training sample set Construct a PC-Kriging model; and perform a PC-Kriging model on the sample set based on the obtained PC-Kriging model. Analyze each sample point x in and get the corresponding response and variance And calculate the estimated failure probability based on the original calculation model Wherein, the PC-Kriging model is: Among them, y=Y(x) is a specific implementation of a Gaussian static random process, and its domain is Y (PCK) (x) represents the PC-Kriging model; is the weighted sum of orthogonal polynomials describing the trend of the PC-Kriging model, α=[α1α2…α M ] T is composed of M non-negative integers α i The vector composed of is the set of all possible α, which is determined by the truncation strategy, y α are the coefficients of the polynomial expansion, is a multivariate polynomial orthogonal to the probability density function corresponding to the input parameters, is the definition of input parameters in chaotic polynomial space; σ 2 is the variance of the Gaussian process, Z(x,ω) is a stationary Gaussian process with zero mean and unit variance, and ω represents the potential probability space; Step S104: using the adaptive point selection strategy, Filter out the best training sample x * ; The adaptive point selection strategy is based on the following learning function: Where p(x) is the joint probability density of random variables at point x; U(x) is the reliability index associated with the probability of misclassification; w pro (x) is the probability weight function; U mod (x_ is the revised reliability index; The selection criteria for the best test points are: Step S105, determine whether the current iteration meets the stopping criteria, if so, go to step S106; otherwise, calculate the best training sample x * The corresponding true response value y * , x * and * Add to Sample Set middle, Then return to step S103; Step S106: Use the test formula to test the sample Is the total number of failures sufficient to accurately estimate the failure probability? If the test formula is satisfied, the final failure probability is output. And get the reliability of the target object; otherwise, it means N MCS The number of candidate design samples is not enough. Increase ΔN MCS candidate design samples, and repeat step S103; the result of the formula is: in, The failure probability The corresponding coefficient of variation, COV thr is the coefficient of variation threshold; The step S2 includes the following steps S201-S210: Step S201, input the initial design point D (0) , set the maximum number of optimization iterations it max , generating the augmented reliability space Step S202, constructing an initial proxy model; using uniform distribution to extract (2n d +1) LHS points, n d is the dimension of the input variable of the corresponding function; these sampling points are used to generate the corresponding proxy model, and the proxy model of the objective function is defined as q=1,2...,k, and the corresponding test sample set is recorded as Surrogate Models for Probabilistic Constraint Functions i=1,2...,n h , and its corresponding test sample set is recorded as Surrogate Models for Deterministic Constraint Functions j=1,2...,n s , and its corresponding test sample set is recorded as Step S203, running the global proxy model enrichment strategy to generate a relatively rough proxy model of the probability constraint function before optimization; adaptively adding sample points with the minimum number of deviations to the corresponding test sample set according to the global proxy model enrichment strategy until the relaxed convergence criterion described by the formula is met; Step S204, initialize the optimization program, set the optimization iteration number s=1, and set the initial training indicator function for all proxy models of the probability constraint function i=1,2...,n h , The proxy model i representing the corresponding performance function needs to be refined locally, and When , there is no need to refine the local proxy model; Step S205, running the optimization program to calculate the RBDO optimization problem, based on the MP criterion combined with the EI criterion combined point adding strategy and the calculation model of the RBDO optimization problem to obtain the sub-optimization problem, based on the current and Solve the sub-optimization problem to approximately obtain the optimal solution or optimal solution set D of the objective function 9s) , and set s=s+1; Step S206, determine whether one of the following two conditions is met, if yes, proceed to step S210, otherwise proceed to step S207; wherein condition 1 is: || D (s) -D (s-1) ||≤1.0×10 -3 and q=1,2...,k; Condition 2: s≥it max ; Among them, D (s) With D (s-1) are two adjacent optimal solutions. and is the value of the qth objective function in the s-1th round and the sth round; Step S207, update the proxy model of the objective function and the deterministic constraint function; for the proxy model of the objective function, if the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than i=1,2...,n h , then add the optimal design point to the training sample set, otherwise, skip the optimal design point, where is the reliability index of the i-th probability constraint; the proxy model for the deterministic constraint function j If the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than i=1,2...,n h , then add the optimal design point as a training point, otherwise, skip the optimal design point; Step S208: Update the training instruction function When the current optimal solution D is determined (s) The optimal solution D of the previous iteration (s -1) The distance in standard normal space is greater than i=1,2...,n h , then set the corresponding The value is 1; Step S209, updating the probability constraint function proxy model; The corresponding proxy model runs the local proxy model refinement strategy, updates the corresponding constraint proxy model, sets s=s+1, and returns to step S205; Step S210, end the program and output the final optimization result.

2. The RBDO method based on the PC-Kriging model and quantile evaluation method according to claim 1, characterized in that: The step S203 specifically includes the following steps: Step S301: uniformly sample a set of candidate design variable samples Where D is a set of design variables including randomness and determinism; Step S302: for each candidate design variable sample Follow these steps: a) Generate N number of s The corresponding sample set Among them, x (k) Design variables for randomness, is an environmental random variable; b) Calculate the sample set The proxy model response c) Calculate the proxy model response set The quantile q α (D (j) ); d) Determine the sample points associated with the quantiles in, For PC-Kriging model The average value of e) Calculate the sample points associated with the quantiles U-function: Among them, u(D (j) ) is a learning function related to quantiles; For U-function, is the failure threshold; is the standard deviation; Step S303: Select all candidate design variable samples The point with the smallest deviation is added as the best improved design point to the experimental sample points of the PC-Kriging model: Step S303, when the number of changes in the sign of the quantile predicted by the PC-Kriging model is small enough in two consecutive enrichment iterations, the algorithm operation is stopped, and it is considered that the PC-Kriging model meets the accuracy requirements at this time; the relaxed convergence criterion here is defined as: Among them, ε sc is the convergence threshold.

3. The RBDO method based on the PC-Kriging model and quantile evaluation method according to claim 1, characterized in that: The step S204 includes the following steps: Step S401, using the Monte Carlo sampling method to generate the design point D corresponding to the current optimization iteration number s (s) Sample set Step S402: Based on the current proxy model Calculate the quantile of the current design point and its corresponding upper and lower boundaries and and Corresponding to and quantile of ; Step S403, determining the training instruction function IND i , i=1,2…,n h Is it equal to 1? i =1, the corresponding agent model i performs subsequent operations accordingly; Step S404, skip the training of the proxy model with non-active constraints; set the number of iterations, and when s is greater than the number of iterations, the training indicator function IND of the non-active constraints is set. i Set to 0; Step S405, add training sample sets and refine the accuracy of the local PC-Kriging proxy model until the local accuracy criterion of the proxy model is met, then IND i Set to 0; the local accuracy criterion of the proxy model is: In the formula, is the preset accuracy threshold; when the failure threshold When Substituting the denominator, where is the PC-Kriging proxy model in the augmented space sample set The variance of Step S406: When all IND i are all set to 0, and the optimization procedure in step S205 is executed; Through the local refinement strategy of the probability constraint surrogate model in the above optimization process, the optimization is combined with the surrogate model enrichment, which solves the remaining epistemic uncertainty problems introduced by the PC-Kriging surrogate model and ensures the accuracy of the evaluation of the quantile of each iterative solution in the optimization process.

4. The RBDO method based on the PC-Kriging model and quantile evaluation method according to claim 3, characterized in that: The step S405 specifically includes: Step S501, referring to the learning function calculation in step S104 Specifically: Among them, u k (D (s) ) is the learning function corresponding to the kth sample; p(·) is the joint probability density of random variables, where the deviation number U can be calculated by the following formula: Step S502: Select the sample set of the current design point in, u k (D (s) ) value is the best improved design point and is added to the experimental sample points of the PC-Kriging model:

5. The RBDO system based on the PC-Kriging model and quantile evaluation method is characterized by: The system comprises: A reliability calculation module, which is used to calculate the reliability of the target object using a reliability analysis algorithm based on an improved adaptive PC-Kriging model; An optimization calculation module, which is used to perform RBDO optimization calculation to obtain an optimization result based on the calculated reliability of the target object; wherein the RBDO optimization calculation includes: constructing a unified proxy model in the entire design domain based on quantiles and augmented reliability space, using a combined point adding strategy of the MP criterion and the EI criterion to make the proxy model approach the optimal solution of the objective function, and introducing a local refinement strategy based on the rough probability constraint proxy model established by the global enrichment strategy so that the probability constraint proxy model can accurately evaluate the quantile of each iterative solution during the optimization process; The reliability calculation module specifically implements the following steps S101-S106: Step S101, based on the random distribution of input parameters, generate N MCS Initial candidate design samples, and these sample sets are recorded as Step S102, from A small number of sample points N0 are randomly selected as the initial training samples for constructing the PC-Kriging model, and the training points are expressed as And calculate the corresponding The true response value Get the training sample set Step S103: based on the training sample set Construct a PC-Kriging model; and perform a PC-Kriging model on the sample set based on the obtained PC-Kriging model. Analyze each sample point x in and get the corresponding response and variance And calculate the estimated failure probability based on the original calculation model Wherein, the PC-Kriging model is: Among them, y=Y(x) is a specific implementation of a Gaussian static random process, and its domain is Y (PCK) (x) represents the PC-Kriging model; is the weighted sum of orthogonal polynomials describing the trend of the PC-Kriging model, α=[α1 α2 … α M ] T is composed of M non-negative integers α i The vector composed of is the set of all possible α, which is determined by the truncation strategy, y α are the coefficients of the polynomial expansion, is a multivariate polynomial orthogonal to the probability density function corresponding to the input parameters, is the definition of input parameters in chaotic polynomial space; σ 2 is the variance of the Gaussian process, Z(x,ω) is a stationary Gaussian process with zero mean and unit variance, and ω represents the potential probability space; Step S104: using the adaptive point selection strategy, Filter out the best training sample x * ; The adaptive point selection strategy is based on the following learning function: Where p(x) is the joint probability density of random variables at point x; U(x) is the reliability index associated with the probability of misclassification; w pro (x) is the probability weight function; U mod (x) is the revised reliability index; The selection criteria for the best test points are: Step S105, determine whether the current iteration meets the stopping criteria, if so, go to step S106; otherwise, calculate the best training sample x * The corresponding true response value y * , x * and * Add to Sample Set middle, Then return to step S103; Step S106: Use the test formula to test the sample Is the total number of failures sufficient to accurately estimate the failure probability? If the test formula is satisfied, the final failure probability is output. And get the reliability of the target object; otherwise, it means N MCS The number of candidate design samples is not enough. Increase ΔN MCS candidate design samples, and repeat step S103; the result of the formula is: in, The failure probability The corresponding coefficient of variation, COV thr is the coefficient of variation threshold; The optimization calculation module specifically implements the following steps S201-S210: Step S201, input the initial design point D (0) , set the maximum number of optimization iterations it max , generating the augmented reliability space Step S202, constructing an initial proxy model; using uniform distribution to extract (2n d +1) LHS points, n d is the dimension of the input variable of the corresponding function; these sampling points are used to generate the corresponding proxy model, and the proxy model of the objective function is defined as q=1,2...,k, and the corresponding test sample set is recorded as Surrogate Models for Probabilistic Constraint Functions i=1,2...,n h , and its corresponding test sample set is recorded as Surrogate Models for Deterministic Constraint Functions j=1,2...,n s , and its corresponding test sample set is recorded as Step S203, running the global proxy model enrichment strategy to generate a relatively rough proxy model of the probability constraint function before optimization; adaptively adding sample points with the minimum number of deviations to the corresponding test sample set according to the global proxy model enrichment strategy until the relaxed convergence criterion described by the formula is met; Step S204, initialize the optimization program, set the optimization iteration number s=1, and set the initial training indicator function for all proxy models of the probability constraint function i=1,2...,n h , The proxy model i representing the corresponding performance function needs to be refined locally, and When , there is no need to refine the local proxy model; Step S205, running the optimization program to calculate the RBDO optimization problem, based on the MP criterion combined with the EI criterion combined point adding strategy and the calculation model of the RBDO optimization problem to obtain the sub-optimization problem, based on the current and Solve the sub-optimization problem to approximately obtain the optimal solution or optimal solution set D of the objective function 9s) , and set s=s+1; Step S206, determine whether one of the following two conditions is met, if yes, proceed to step S210, otherwise proceed to step S207; wherein condition 1 is: || D (s) -D (s-1) ||≤1.0×10 -3 and q=1,2...,k; Condition 2: s≥it max ; Among them, D (s) With D (s-1) are two adjacent optimal solutions. and is the value of the qth objective function in the s-1th round and the sth round; Step S207, update the proxy model of the objective function and the deterministic constraint function; for the proxy model of the objective function, if the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than i=1,2...,n h , then add the optimal design point to the training sample set, otherwise, skip the optimal design point, where is the reliability index of the i-th probability constraint; the proxy model for the deterministic constraint function j If the current optimal solution D (s) arrive The minimum distance between the points in the standard normal space is greater than i=1,2...,n h , then add the optimal design point as a training point, otherwise, skip the optimal design point; Step S208: Update the training instruction function When the current optimal solution D is determined (s) The optimal solution D of the previous iteration (s -1) The distance in standard normal space is greater than i=1,2...,n h , then set the corresponding The value is 1; Step S209, updating the probability constraint function proxy model; The corresponding proxy model runs the local proxy model refinement strategy, updates the corresponding constraint proxy model, sets s=s+1, and returns to step S205; Step S210, end the program and output the final optimization result.

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