Reliability optimization design method for complex equipment based on random agent model

Abstract

The invention discloses a complex equipment reliability optimization design method based on a random agent model, belongs to the field of complex equipment reliability optimization design, and effectively constructs a black box agent model containing the influence of unknown and uncontrollable input random parameters on the response of complex equipment. And reliability evaluation is carried out on the complex equipment component by combining a Monte Carlo method, and finally reliability optimization design is realized. The invention provides a complex equipment reliability optimization design method considering implicit input parameter uncertainty, which comprises the following steps of: representing a random field of random response of a complex equipment part through Carloam-Roseli expansion, and agenting the represented random field by combining polynomial chaos expansion and a kernel density estimation method; therefore, the random behavior reproduction of the complex equipment part response under the implicit random parameter influence is realized. The method can be used for searching the optimal design under the influence of unknown and uncontrollable random input parameters in the performance response of complex equipment.

Description

Technical Field

[0001] The present invention belongs to the field of complex equipment reliability optimization design, and in particular relates to a complex equipment reliability optimization design method based on a random agent model. Background Art

[0002] In the field of engineering, especially in the optimization design process of complex equipment, surrogate models are widely used as an important tool to improve design efficiency and reduce computational costs. The main function of the surrogate model is to reduce the amount of calculation required when directly calling the high computational cost simulation model by approximating the complex simulation model. However, the surrogate models currently used in most complex equipment optimization design methods are deterministic models, that is, under the same input conditions, repeated calls to the model can always get a consistent response. Such models usually assume that all input variables are deterministic and the output response of the system is a unique value.

[0003] However, in actual engineering applications, especially in the design process of complex equipment, system responses often show great uncertainty. When optimizing the design of complex equipment through professional simulation software, even if the input conditions remain consistent, the output response (such as certain key performance indicators, load response, etc.) is not unique. This is because there are not only explicit input random variables (such as design size, pressure, load, etc.) in the system, but also many implicit random factors (such as material properties, environmental fluctuations, etc.), which jointly affect the randomness of the system output. Therefore, relying solely on deterministic proxy models cannot fully capture and describe the true behavior of complex equipment under actual working conditions.

[0004] At present, although the impact of explicit input random variables has been widely studied, in complex systems, many implicit random factors also have a significant impact on system response. These implicit factors are usually difficult to quantify directly, but their contribution to performance uncertainty cannot be ignored. Existing uncertainty quantification methods usually require a lot of computing resources and multiple complex simulations to evaluate the uncertainty of the output response, which is not only computationally expensive but also difficult to implement in actual engineering applications. Therefore, how to more efficiently and accurately quantify the uncertainty of complex equipment responses and optimize the design has become a major challenge in current reliability optimization design. Summary of the invention

[0005] In order to solve the shortcomings of the prior art and effectively consider the impact of the uncertainty of the service environment of complex equipment on the uncertainty of component performance response, so as to avoid design deficiencies or redundancies, the present invention adopts the following technical solutions:

[0006] The reliability optimization design method of complex equipment based on random agent model includes the following steps:

[0007] Step S1: determine the initial training sample points; add the initial training sample points in sequence under the same response surface, and use the polynomial chaos expansion PCE to construct the PCE model of the response surface until the PCE model accuracy converges, and obtain the initial training sample points of the final PCE model;

[0008] Step S2: construct a random proxy model; first characterize the random field of complex equipment performance response, then construct a random proxy model of performance response through the coefficients of the proxy random field, and continuously simulate sampling at the initial training sample point until the accuracy of the random proxy model converges;

[0009] Step S3: setting the initial solution, i.e. the initial design to be optimized, and setting the iteration conditions;

[0010] Step S4: Perform reliability analysis based on the constructed performance response random proxy model;

[0011] Step S5: Determine whether the optimal solution converges. If the convergence condition is met, output the optimal solution, otherwise return to step S4.

[0012] Furthermore, in step S1, initial training sample points are generated, and the Sobol sampling method is used to sample and the following formula is used to judge the accuracy convergence of the proxy model:

[0013]

[0014] in, Indicates The response prediction error of the constrained surrogate model, Indicates the sample point of the last Sobol sampling, express On the response surface Next A true constraint response, express On the response surface Next The constrained PCE model, represents a known and controllable explicit input random variable, express On the response surface Next A true constraint response, It represents the difference between the maximum response and the minimum response in the training sample points of Sobol sampling, which is used to normalize the response error. represents the error threshold, which is set to In order to ensure the global accuracy of the PCE model and the uniformity of the distribution of sample points in the design space with as few samples as possible, when the accuracy of the PCE model converges, twice the number of training sample points are sequentially extracted from the Sobol sampling sequence to form the initial training sample points of the final PCE model. .

[0015] Furthermore, in step S2, the Karhunen–Loève expansion is used to characterize the performance response random field, and the formula is as follows:

[0016]

[0017] in, represents a known and controllable explicit input random variable, represents unknown and uncontrollable implicit input random parameters (which can also be viewed as different response surfaces), represents the random response model of the random field, Represents a set of random field response mean functions, that is, random variables of n dimensions The mean of A set of random numbers representing the first dimension of design variables, A set of random numbers representing the second dimension of design variables, A set of random numbers representing the n-th dimension design variable, k is an integer from 1 to infinity, represents the corresponding eigenvalue, represents a set of independent random numbers with mean zero, represents a set of orthonormal bases, The covariance function of the random field is given by The points are:

[0018]

[0019] ,

[0020] in, Indicates except There is another group (or another dimension) of random variables, E[·] is the expectation operator, which means the expectation of the calculation results in [].

[0021] Furthermore, in step S2, the performance response random proxy model constructed based on the random field represented by Karhunen–Loève includes the proxy response mean function ;

[0022] Use the same input initial training sample points Simulation response samples obtained under different response surfaces train PCE models:

[0023]

[0024] in, represents the first determined response surface, represents the second determined response surface, represents the Rth determined response surface, represents the PCE proxy model trained with the rth group of initial training sample point data, represents the coefficients of the r-th PCE model calculated using least angle regression (LARS), and the PCE coefficients are adaptively truncated using q-norm. represents the orthogonal basis of PCE;

[0025] Use this The PCE model constructs the response mean function by the following formula The proxy response mean function :

[0026] .

[0027] Furthermore, in step S2, the performance response random proxy model constructed based on the random field characterized by Karhunen–Loève includes the proxy eigenvalue and orthonormal basis ;

[0028] The orthogonal basis is calculated by integrating the covariance function and eigenvalues :

[0029]

[0030] in, represents the covariance function of the agent, Represents the input random variable The probability density function of The spectrum is expressed as , the covariance function of the agent is expressed as:

[0031]

[0032] After integrating the covariance function, we get a set of eigenvalues and orthogonal basis coefficients , orthogonal basis coefficients pass Get the orthogonal basis , and finally obtain a set of eigenvalues and orthonormal basis , since k has infinite terms, the obtained eigenvalues ​​and orthogonal bases are truncated, and the truncation formula is:

[0033]

[0034] Here, K means truncation at K.

[0035] Furthermore, in step S2, the performance response random proxy model constructed based on the random field represented by Karhunen–Loève includes a proxy random number The statistical distribution of

[0036] The load response obtained by the simulation sample is projected into the orthogonal basis space, which is achieved by the following formula:

[0037]

[0038] Based on projection to orthogonal basis space Group random number , fit the probability density distribution of this set of data through kernel density estimation (KDE) or Copula function , so that the random number in the random agent model Agent of.

[0039] Furthermore, in step S2, the random field represented by the proxy Karhunen–Loève expansion includes the response mean function , eigenvalue and orthonormal basis And the statistical distribution of random numbers , a random proxy model of complex equipment performance response under the combined effect of explicit input random variables and implicit input random parameters was constructed , the specific formula is:

[0040]

[0041] in, Represents the probability density distribution The random number obtained by random sampling is Agents;

[0042] Constantly at the initial training sample point Simulation sampling refers to sampling on the same response surface. Next, input sample points Simulate sampling to obtain a set of performance response samples , sample R groups of simulation samples under different response surfaces , by using the following convergence criteria to determine the number of training samples in group R:

[0043]

[0044] in, represents the random proxy model error of the jth constraint, T represents the number of training sample points, Represents the second-order Watsonstein distance, which is used to measure the error between the output distributions of two models. Represents the standard deviation of the response output by the true random model M(x) at the current training sample point, Represents the random surrogate model error threshold.

[0045] Furthermore, in step S4, the constructed performance response random proxy model is used to calculate the failure probability through Monte Carlo simulation MCS.

[0046] Furthermore, in step S5, the sensitivity of the failure probability to the design variables is calculated using a scoring function, which is specifically expressed as follows:

[0047]

[0048] in, Indicates the first Constraints The failure probability, Indicates The mean of the design variables, represents the total number of samples used in the MCS simulation to calculate the failure probability, n represents the sample index, It represents the indicator function for judging whether a sample is a failure sample, which is specifically expressed as:

[0049]

[0050] in, Indicates the failure domain, represents the scoring function.

[0051] Furthermore, for design variables that obey different statistical distributions, the scoring function Different, including normal distribution, lognormal distribution, Gumbel distribution, Weibull distribution:

[0052] normal distribution:

[0053] Lognormal distribution:

[0054] Gumbel Distribution:

[0055] Weibull Distribution:

[0056] The convergence index of the optimal solution specifically refers to the following formula, which is used to judge whether the optimal solution of the optimization design process converges:

[0057] .

[0058] The advantages and beneficial effects of the present invention are:

[0059] In view of the problem that the service environment of complex equipment has strong uncertainty and the performance response is difficult to accurately characterize, the method of the present invention combines the Karhunen–Loève expansion, PCE model and KDE method to construct a random proxy model of the performance response of complex equipment. Unlike most existing proxy models, this random proxy model can mine the random behavior of performance response based on training sample data, and at the same time can output random performance responses based on statistical reasoning under the same input variables. Finally, the present invention uses the random proxy model to perform reliability optimization design on complex equipment, which can effectively consider the impact of the uncertainty of the service environment of complex equipment on the uncertainty of component performance response, thereby overcoming the problems of insufficient or redundancy in existing design methods and achieving a more economical and reliable optimal design. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a flow chart of a method according to an embodiment of the present invention.

[0061] Figure 2 This is a diagram showing the training effect of the random proxy model with constraint G1 in the first embodiment of the present invention.

[0062] Figure 3 This is a diagram showing the training effect of the random proxy model with constraint G2 in the first embodiment of the present invention.

[0063] Figure 4 This is a diagram showing the training effect of the random proxy model with constraint G3 in the first embodiment of the present invention.

[0064] Figure 5 Schematic diagram of wind turbine tower model parameters and stress conditions in the second embodiment of the present invention.

[0065] Figure 6 is the extreme load of the wind turbine tower (tower top bending moment) in the second embodiment of the present invention. )’s random proxy model training effect diagram.

[0066] Figure 7 The wind turbine tower extreme load (gravity load at the bottom of the tower) in the second embodiment of the present invention )’s random proxy model training effect diagram.

[0067] Figure 8 is the extreme load of the wind turbine tower (bending moment at the bottom of the tower) in the second embodiment of the present invention. )’s random proxy model training effect diagram. DETAILED DESCRIPTION

[0068] The specific implementation of the present invention is described in detail below in conjunction with the accompanying drawings. It should be understood that the specific implementation described here is only used to illustrate and explain the present invention, and is not used to limit the present invention.

[0069] The reliability optimization design method of complex equipment based on random agent model of the present invention, under the condition that the uncertainty of complex equipment performance response is caused by implicit input random parameters, combines KL (Karhunen–Loève) expansion, PCE (Polynomial Chaos Expansions) model, KDE (Kernel Density Estimation) method, constructs a random agent model of complex equipment performance response, and uses MCS (Monte Carlo Simulation) to perform reliability analysis. Finally, the sensitivity of failure probability to design variables is calculated using a scoring function, and SQP (Sequential Quadratic Programming) is used to perform optimization design, such as Figure 1 As shown, the specific steps include:

[0070] Step S1: Determine the initial training sample points ; On the same response surface (using represents a certain response surface) and then increases the initial training sample points , and use the polynomial chaos expansion (PCE) to build the PCE model of the response surface until the accuracy of the PCE model converges;

[0071] Generate initial training sample points, use Sobol sampling method to sample, and use the following formula to judge the accuracy convergence of the proxy model:

[0072]

[0073] in, Indicates The response prediction error of the constrained surrogate model, Indicates the sample point of the last Sobol sampling, express On the response surface Next A true constraint response, express On the response surface Next The constrained PCE model, represents a known and controllable explicit input random variable, express On the response surface Next A true constraint response, It represents the difference between the maximum response and the minimum response in the training sample points of Sobol sampling, which is used to normalize the response error. represents the error threshold, which is set to In order to ensure the global accuracy of the PCE model and the uniformity of the distribution of sample points in the design space with as few samples as possible, when the accuracy of the PCE model converges, twice the number of training sample points are sequentially extracted from the Sobol sampling sequence to form the initial training sample points of the final PCE model. .

[0074] Step S2: Constructing a random proxy model ; Use the Karhunen–Loève expansion to characterize the random field of complex equipment performance response, and construct a random proxy model of performance response by proxying the coefficients of the random field, and continuously train the initial training sample points The simulation is sampled until the accuracy of the random proxy model converges.

[0075] The Karhunen–Loève expansion is used to characterize the performance response random field, and the formula is as follows:

[0076]

[0077] in, represents a known and controllable explicit input random variable, represents unknown and uncontrollable implicit input random parameters (which can also be viewed as different response surfaces), represents the random response model of the random field, Represents the mean function of the random field response, that is, a random variable of n dimensions The mean of A set of random numbers representing the first dimension of design variables, A set of random numbers representing the second dimension of design variables, A set of random numbers representing the n-th dimension design variable, represents the corresponding eigenvalue, represents a set of independent random numbers with mean zero, represents a set of orthonormal bases, The covariance function of the random field is given by The points are:

[0078]

[0079] ,

[0080] in, Indicates except There is another group (or another dimension) of random variables, E[·] is the expectation operator, which means the expectation of the calculation results in [].

[0081] The performance response random proxy model is constructed based on the random field represented by the Karhunen–Loève expansion, and the mean function of the proxy response is sequentially , eigenvalue and orthonormal basis , random number The statistical distribution of is:

[0082] 1) Agent response mean function :

[0083] Use the same input initial training sample points Simulation response samples obtained under different response surfaces train PCE models:

[0084]

[0085] in, represents the second determined response surface, represents the Rth determined response surface, represents the PCE proxy model trained with the rth group of initial training sample point data, represents the coefficients of the r-th PCE model calculated using least angle regression (LARS), and the PCE coefficients are adaptively truncated using q-norm. represents the orthogonal basis of PCE.

[0086] Use this The PCE model constructs the response mean function by the following formula The proxy response mean function :

[0087]

[0088] 2) Proxy feature value and orthonormal basis :

[0089] The orthogonal basis is calculated by integrating the covariance function and eigenvalues :

[0090]

[0091] in, represents the covariance function of the agent, Represents the input random variable The probability density function of The spectrum is expressed as , the covariance function of the agent is expressed as:

[0092]

[0093] After integrating the covariance function, we can get a set of eigenvalues and orthogonal basis coefficients , orthogonal basis coefficients pass Get the orthogonal basis , and finally obtain a set of eigenvalues and orthonormal basis Since k has infinite terms, the obtained eigenvalues ​​and orthogonal bases need to be truncated at K. The truncation formula is:

[0094]

[0095] 3) Proxy random number Statistical distribution of:

[0096] The load response obtained by the simulation sample is projected into the orthogonal basis space, which is achieved by the following formula:

[0097]

[0098] Based on projection to orthogonal basis space Group random number , fit the probability density distribution of this set of data through kernel density estimation (KDE) or Copula function , thereby realizing the random number in the random agent model Agent of.

[0099] Random fields characterized by surrogate Karhunen–Loève expansion, including response mean function , eigenvalue and orthonormal basis And the statistical distribution of random numbers , a random proxy model of complex equipment performance response under the combined effect of explicit input random variables and implicit input random parameters was constructed , the specific formula is:

[0100]

[0101] in, Represents the probability density distribution The random number obtained by random sampling is Agent of.

[0102] Constantly at the initial training sample point Simulation sampling refers to sampling on the same response surface. Next, input sample points The simulation sampling obtains a set of performance response samples, denoted as , sample R groups of simulation samples under different response surfaces , by using the following convergence criteria to determine the number of training samples in group R:

[0103]

[0104] in, represents the random proxy model error of the jth constraint, T represents the number of training sample points, Represents the second-order Wasserstein distance, which is used to measure the error between the output distributions of two models. Represents the standard deviation of the response output by the true random model M(x) at the current training sample point, represents the random surrogate model error threshold, which is set to .

[0105] Step S3: Set the initial solution , that is, the initial design, usually the design variable that needs to be optimized before the optimization starts, x represents the design variable, and sets the number of iterations .

[0106] Step S4: Based on the constructed performance response random agent model , reliability analysis is performed. As an implementation, the failure probability is calculated by Monte Carlo simulation (MCS) using the constructed performance response random surrogate model.

[0107] Step S5: Determine whether the optimal solution converges. If the convergence condition is met, output the optimal solution, otherwise return to step S4.

[0108] The sensitivity of failure probability to design variables is calculated using a scoring function, which is expressed as follows:

[0109]

[0110] in, Indicates the first Constraints The failure probability, Indicates The mean of the design variables, represents the total number of samples used by MCS to calculate the failure probability, n represents the sample index, It represents the indicator function for judging whether a sample is a failure sample, which is specifically expressed as:

[0111]

[0112] in, Indicates the failure domain, Represents the scoring function. For design variables that obey different statistical distributions, the scoring function will be different, as shown below:

[0113] normal distribution:

[0114] Lognormal distribution:

[0115] Gumbel Distribution:

[0116] Weibull Distribution:

[0117] The convergence index of the optimal solution specifically refers to the following formula, which is used to judge whether the optimal solution of the optimization design process converges:

[0118]

[0119] Embodiment 1:

[0120] In order to verify the effectiveness of the method of the present invention, this implementation case adopts the model and data given by Wu (Wu Z, Chen Z, Chen G, et al. A probability feasible region enhanced important boundary sampling method for reliability-based design optimization [J]. Structural and Multidisciplinary Optimization, 2021, 63: 341-355.) and substitutes them into the method of the present invention. In order to simulate the potential randomness in actual engineering in a computer, the present invention gives randomness to some parameters of the model. This embodiment consists of two independent normally distributed random design variables and three constraint functions. The design variable is the mean of the random variable. , where the implicit parameter It is set as unknown to simulate random factors that are not or cannot be considered in actual engineering. The random response caused by implicit parameters cannot be eliminated artificially in the process of reliability analysis and optimization design. The optimization problem is defined as:

[0121]

[0122] In the formula, represents the failure probability operator, represents the standard normal cumulative distribution function operator, Represents the target reliability corresponding to the target failure probability.

[0123] In this example, the PCE model was constructed using the open source code UQ lab, the PCE coefficients were calculated using LARS, and the PCE was adaptively truncated using the q-norm method. First, a 2D Sobol sequence sample was generated based on the 2D input variable, and the Sobol samples were added sequentially to train the PCE model under a response surface. The training samples were determined based on the convergence criterion. Then, in the training sample The PCE model under different response surfaces is constructed, and the random proxy model with corresponding constraints is constructed. Since the random response of the random proxy model causes slight fluctuations in the convergence index, in order to ensure the continuity of the convergence of the random proxy model, the batch training method is used to check the convergence of the random proxy model, that is, 50 PCE models are used as a batch to train the random proxy model, and the random proxy model is trained in batches until convergence. Finally, the converged random proxy model is used to perform reliability optimization design. Each reliability analysis is performed using MCS was performed on each sample.

[0124] The random proxy model trained by the method of the present invention can better reproduce the randomness of the constraint response under implicit random parameters, such as Figures 2 to 4 The results obtained by the method of the present invention are compared with the results without considering the implicit random parameters and the results with considering the implicit random parameters using the explicit function, as shown in Table 1. The reliability optimization design results without considering the implicit random parameters seriously fail to meet the target reliability. The reliability optimization design results using explicit functions to consider implicit random parameters meet the target reliability. However, it is difficult to meet the requirements of actual engineering optimization by obtaining tens of thousands of performance responses. In contrast, the method of using a random proxy model to consider implicit random parameters (this method) only requires a relatively small number of performance responses to meet the target reliability through optimization. The optimal solution of reliability optimization design is required. This is mainly because the method of the present invention uses the PCE model and kernel density estimation (or copula function) based on the Karhunen–Loève expansion and some training samples to perform global statistical inference on the randomness of the random field, and performs MCS based on the statistical characteristics of the inference, rather than blindly performing MCS directly, which results in a large amount of performance response consumption.

[0125] Table 1 Comparison of calculation results of implementation case 1

[0126]

[0127] Embodiment 2:

[0128] The NREL 5MW wind turbine tower optimization case under the 1.3 working condition (turbulent wind working condition) specified in the IEC 61400-9 standard is used to test the effectiveness of the proposed method for engineering application problems. The wind turbine tower optimization model uses the tower volume (steel quantity) as the objective function, and the constraint function is composed of the ultimate stress and the tower top torsion angle. The tower model parameters and stress conditions are as follows: Figure 5 and as shown in Table 2. The optimization problem is defined as:

[0129]

[0130] Table 2 5MW tower model parameters

[0131]

[0132] The tower model is simplified to a cone segment model formed by linear interpolation of tower bottom size parameters and tower top size parameters. and are the diameters of the tower base and top, respectively, which follow a normal distribution with a standard deviation of 0.06, 004; is the ultimate stress constraint of the tower; is the allowable ultimate stress value, is the gravity load at the bottom of the tower, is the cross-sectional area of ​​the tower bottom, is the bending moment at the tower base, is the bending moment at the top of the tower, is the section bending modulus, is the torsion angle constraint of the tower top. Here, the allowable value of the ultimate torsion angle of the tower top is set to 5°. The failure probabilities of the ultimate stress constraint and the torsion angle constraint of the tower top must both meet the target reliability. The load response in the tower constraint is generated by TurbSim to generate a turbulent wind file, which is then simulated by Openfast.

[0133] As an ultimate constraint, the IEC 61400 standard stipulates that the load extreme value of the 10-minute simulation is used as the ultimate load response. Under turbulent wind conditions, even with the same simulation input settings, the existence of the random wind seed will cause the simulation extreme load result to be a random number. Because the random seed is not a valid parameter for measuring the random load distribution, the extreme load in Implementation Case 2 The randomness of is difficult to quantify accurately and is an implicit random parameter in the optimization model.

[0134] In the process of training the random proxy model, the statistical distribution of the 10-minute average wind speed and the statistical distribution of the tower load response are unknown. Openfast simulation can obtain both The random proxy model trained by the method of the present invention using 1200 sets of simulation data is compared with 5440 sets of openfast simulation samples. The random proxy model can better reproduce the randomness of extreme value loads, as shown in the following figure. Figures 6 to 8 shown.

[0135] The results obtained by the method of the present invention are compared with the initial design of the tower under the working condition 1.3, and the results are shown in Table 3. With only a small number of simulations, the method of the present invention obtains more accurate reliability optimization results, which shows that the method has made contributions to solving the reliability optimization design containing implicit random parameters (random factors that are not considered or cannot be accurately considered).

[0136] Table 3 Summary of wind turbine tower optimization results

[0137]

[0138] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some or all of the technical features thereof may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A reliability optimization design method for complex equipment based on a random agent model, characterized by The steps include: Step S1: determine the initial training sample points; add the initial training sample points in sequence under the same response surface, and use the polynomial chaos expansion PCE to construct the PCE model of the response surface until the PCE model accuracy converges, and obtain the initial training sample points of the final PCE model; Step S2: construct a random proxy model; first characterize the random field of complex equipment performance response, then construct a random proxy model of performance response through the coefficients of the proxy random field, and continuously simulate sampling at the initial training sample point until the accuracy of the random proxy model converges; Step S3: setting the initial solution, i.e. the initial design to be optimized, and setting the iteration conditions; Step S4: Perform reliability analysis based on the constructed performance response random proxy model; Step S5: Determine whether the optimal solution converges. If the convergence condition is met, output the optimal solution, otherwise return to step S4.

2. The complex equipment reliability optimization design method based on stochastic agent model according to claim 1 is characterized by: In step S1, initial training sample points are generated, and the accuracy convergence of the proxy model is judged by sampling and using the following formula: ， in, Indicates The response prediction error of the constrained surrogate model, represents the last sampling point, express On the response surface Next A true constraint response, express On the response surface Next The constrained PCE model, represents a known and controllable explicit input random variable, express On the response surface Next A true constraint response, Represents the difference between the maximum response and the minimum response in the sampled training sample points, Represents the error threshold. When the PCE model converges, 2 times the training sample points are extracted from the sampling sequence to form the initial training sample points of the final PCE model. .

3. The complex equipment reliability optimization design method based on stochastic agent model according to claim 1 is characterized by: In step S2, the Kallenin-Loewe expansion is used to characterize the performance response random field, and the formula is as follows: ， in, represents a known and controllable explicit input random variable, represents unknown and uncontrollable implicit input random parameters, represents the random response model of the random field, represents a set of random field response mean functions, k is an integer from 1 to infinity, represents the corresponding eigenvalue, represents a set of independent random numbers with mean zero, represents a set of orthonormal bases, The covariance function of the random field is given by The points are: ， ， in, Indicates except is another random variable, and E[·] is the expectation operator.

4. The complex equipment reliability optimization design method based on stochastic agent model according to claim 3 is characterized by: In step S2, the performance response random proxy model constructed based on the random field of Kallenin-Loy characterization includes the proxy response mean function ; Use the same input initial training sample points Simulation response samples obtained under different response surfaces train PCE models: ， in, represents the first determined response surface, represents the second determined response surface, represents the Rth determined response surface, represents the PCE proxy model trained with the rth group of initial training sample point data, represents the coefficients of the r-th PCE model calculated using least angle regression, represents the orthogonal basis of PCE; Use this The PCE model constructs the response mean function by the following formula The proxy response mean function : 。 5. The complex equipment reliability optimization design method based on stochastic agent model according to claim 3 is characterized by: In step S2, the performance response random agent model constructed based on the random field of Kallenin-Loy characterization includes the agent eigenvalue and orthonormal basis ; The orthogonal basis is calculated by integrating the covariance function and eigenvalues : ， in, represents the covariance function of the agent, Represents the input random variable The probability density function of The spectrum is expressed as , the covariance function of the agent is expressed as: ， After integrating the covariance function, we get a set of eigenvalues and orthogonal basis coefficients , orthogonal basis coefficients pass Get the orthogonal basis , and finally obtain a set of eigenvalues and orthogonal basis , since k has infinite terms, the obtained eigenvalues ​​and orthogonal bases are truncated.

6. The complex equipment reliability optimization design method based on stochastic agent model according to claim 1 is characterized by: In step S2, the performance response random proxy model constructed based on the random field of Kallenin-Loy characterization includes a proxy random number The statistical distribution of The load response obtained by the simulation sample is projected into the orthogonal basis space, which is achieved by the following formula: ， Based on projection to orthogonal basis space Group random number , by fitting the probability density distribution of this set of data , so that the random number in the random agent model Agent of.

7. The complex equipment reliability optimization design method based on stochastic agent model according to claim 3 is characterized by: In step S2, the random field represented by the proxy Karolinen-Loewy expansion includes the response mean function , eigenvalue and orthogonal basis And the statistical distribution of random numbers , a random proxy model of complex equipment performance response under the combined effect of explicit input random variables and implicit input random parameters was constructed , the specific formula is: ， in, Represents the probability density distribution The random number obtained by random sampling is Agents; Constantly at the initial training sample point Simulation sampling refers to sampling on the same response surface. Next, input sample points Simulate sampling to obtain a set of performance response samples , sample R groups of simulation samples under different response surfaces , by using the following convergence criteria to determine the number of training samples in group R: ， in, represents the random proxy model error of the jth constraint, T represents the number of training sample points, represents the second-order Watsonstein distance, which is used to measure the error between the output distributions of two models. Represents the standard deviation of the response output by the true random model M(x) at the current training sample point, Represents the random surrogate model error threshold.

8. The complex equipment reliability optimization design method based on stochastic agent model according to claim 1 is characterized by: In step S4, the failure probability is calculated by Monte Carlo simulation using the constructed performance response random proxy model.

9. The complex equipment reliability optimization design method based on stochastic agent model according to claim 1 is characterized by: In step S5, the sensitivity of the failure probability to the design variables is calculated using a scoring function, which is specifically expressed as follows: ， Where x represents a random input sample, The simulation calculation Constraints The failure probability, Indicates The means of the design variables, represents the total number of samples used in the simulation to calculate the failure probability, n represents the sample index, It represents the indicator function for judging whether a sample is a failure sample, which is specifically expressed as: ， in, Indicates the failure domain, represents the scoring function.

10. The complex equipment reliability optimization design method based on stochastic agent model according to claim 9 is characterized by: The scoring function is different for design variables that obey different statistical distributions, including normal distribution, lognormal distribution, Gumbel distribution, and Weibull distribution.

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