Method for analyzing sensitivity of distribution parameter uncertainty of radar cover structure reliability

Abstract

The present disclosure relates to a method for analyzing sensitivity of distribution parameter uncertainty of a radome structure, and relates to the field of uncertainty analysis, which comprises: constructing an adaptive Kriging surrogate model based on a first initial training sample set; calculating a total variance value and a first conditional variance value of an output failure probability of a real radome structure model according to the adaptive Kriging surrogate model, and calculating a main sensitivity of the real radome structure model according to the total variance value and the first conditional variance value; calculating a second conditional variance value of the output failure probability according to the adaptive Kriging surrogate model, and calculating a conditional expectation value of the output failure probability according to the second conditional variance value; calculating a total sensitivity of the real radome structure model according to the second conditional variance value and the conditional expectation value, and analyzing the sensitivity of the real radome structure model according to the main sensitivity and the total sensitivity. The present disclosure improves the sensitivity analysis efficiency.

Description

Technical Field

[0001] This disclosure relates to the field of uncertainty analysis, and more specifically, to a method for analyzing the reliability and sensitivity of distributed parameter uncertainty of a radome structure. Background Technology

[0002] In harsh environments, aircraft radar antennas can be damaged by lightning strikes, hail, and other environmental factors, leading to failure and even more serious flight accidents. The radome is a crucial structural component of an aircraft, protecting the radar antenna system and providing safety. Therefore, the performance of the radome directly affects the normal operation of the radar system. Uncertainties in the material properties and geometric parameters of the radome structure have a significant impact on its output performance. Therefore, assessing the propagation of these uncertainties and their impact is necessary, and sensitivity analysis is a key method for achieving this.

[0003] The aforementioned uncertainties generally fall into two categories: objective uncertainties (such as the uncertainty of input variables) and subjective uncertainties (such as the uncertainty of distribution parameters). Due to a lack of understanding of the structure and insufficient information gathering, distribution parameters exhibit subjective uncertainty. Therefore, when considering the uncertainty of the distribution parameters of a radome structure, global reliability sensitivity analysis can measure the individual effect of the distribution parameters of the input variables as they vary across their entire uncertainty range, as well as their average contribution to the output failure probability through interactions with other distribution parameter variables. Furthermore, it provides a ranking of the importance of each distribution parameter, aiming to offer guidance on reducing the impact of uncertainty on structural output failure while ensuring the wider application of radome structures.

[0004] When considering the uncertainty of distributed parameters, the Monte Carlo method is a fundamental approach in digital simulation for global reliability sensitivity analysis. However, its computational complexity makes it unacceptable for solving engineering problems. Therefore, to reduce the computational burden of reliability sensitivity analysis in engineering structures, some solutions simplify the process of transferring the uncertainty of distributed parameters to failure probabilities into a "black box" model, defining a global reliability sensitivity index for the distributed parameters and using the Kriging method to reduce the computational load in the index calculation. However, for radome structures, due to their complexity and high dimensionality of variables, how to model the radome structure and efficiently analyze its sensitivity based on this model remains a critical problem to solve.

[0005] It should be noted that the information in the background section above is only used to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0006] The purpose of this disclosure is to provide a method for analyzing the reliability sensitivity of distributed parameters of a radome structure, thereby overcoming, to at least some extent, the problem of inefficient sensitivity analysis of radome structures due to limitations and defects in related technologies.

[0007] According to one aspect of this disclosure, a method for reliability and sensitivity analysis of distributed parameter uncertainty of a radome structure is provided, comprising:

[0008] The first initial training sample set is constructed based on the original distribution parameters of the real radome structure model, and an adaptive Kriging proxy model is constructed based on the first initial training sample set.

[0009] The total variance and the first conditional variance of the output failure probability of the real radome structure model are calculated based on the adaptive Kriging surrogate model, and the main sensitivity of the real radome structure model is calculated based on the total variance and the first conditional variance.

[0010] The second conditional variance of the output failure probability is calculated based on the adaptive Kriging surrogate model, and the conditional expected value of the output failure probability is calculated based on the second conditional variance.

[0011] The total sensitivity of the real radome structure model is calculated based on the second conditional variance value and the conditional expectation value, and the sensitivity of the real radome structure model is analyzed based on the main sensitivity and the total sensitivity.

[0012] In one exemplary embodiment of this disclosure, a first initial training sample set is constructed based on the original distribution parameters of a real radome structure model, including:

[0013] Obtain the original distribution parameters of the real radome structure model, and use the first target proxy importance sampling probability density function to sample the original input variables to obtain multiple sampling results;

[0014] A first sample pool is constructed based on the multiple sampling results, and multiple first original sample points are randomly drawn from the first sample pool.

[0015] Calculate the first distance between the first original sample point and the origin of the real radome structure model, and select multiple first target sample points from the first original sample points based on the first distance;

[0016] The first buckling response value of each of the first target sample points is calculated using the original model function of the real radome structure model, and the first initial training sample set is constructed based on the first buckling response value.

[0017] In one exemplary embodiment of this disclosure, the original input variable is sampled using a first target surrogate importance sampling probability density function to obtain multiple sampling results, including:

[0018] Based on the selection principle of the surrogate sampling probability density function, the first original surrogate sampling probability density function of the real radome structure model is determined;

[0019] In an independent standard normal space, the design points and reliability indices of the real radome structure model are calculated using the improved first-order second-moment method.

[0020] The first original agent sampling probability density function is shifted from the sampling center to the design point to obtain the first target agent importance sampling probability density function.

[0021] The original input variables are sampled using the first target proxy importance sampling probability density function to obtain multiple sampling results.

[0022] In one exemplary embodiment of this disclosure, constructing an adaptive Kriging proxy model based on a first initial training sample set includes:

[0023] A target function is constructed based on the original model function of the real radome structure model and the preset failure conditions, and the initial Kriging proxy model is constructed based on the target function and the first initial training sample set.

[0024] The U-learning function value of the sampling results included in the first sample pool is calculated using the initial Kriging proxy model, and sample points to be updated are selected from the sampling results in the first sample pool based on the U-learning function value.

[0025] When the U-learning function value of the sample point to be updated is determined to be greater than or equal to a preset threshold, the initial Kriging proxy model is used as the adaptive Kriging proxy model.

[0026] In one exemplary embodiment of this disclosure, calculating the total variance of the output failure probability of the real radome structure model based on an adaptive Kriging surrogate model includes:

[0027] Calculate the first multidimensional distribution parameter variable of the original distribution parameter in an independent standard normal space, and calculate the first variable integration point and the first weight value of the first multidimensional distribution parameter variable;

[0028] The independent standard normal space is transformed using the integration point of the first variable to obtain the first original model space, and multiple sampling results are selected from the first sample pool to construct the first input variable sample under the first original model space;

[0029] The first failure domain indicator function is calculated based on the adaptive Kriging proxy model and the first input variable sample. The first failure probability integral point of the output failure probability of the real radome structure model is calculated based on the first target proxy importance sampling probability density function, the first subjective probability density function, the first β-sphere indicator function, the first failure domain indicator function, and the first variable integral point of the first input variable sample.

[0030] The total variance of the output failure probability is calculated based on the first failure probability integral point and the first weight value.

[0031] In one exemplary embodiment of this disclosure, calculating the first conditional variance of the output failure probability of the real radome structure model based on an adaptive Kriging surrogate model includes:

[0032] In the first original model space, calculate the second variable integration point and the second weight value of the second multidimensional distribution parameter variable included in the first original model space, and transform the second variable integration point to the first original model space to obtain the second original model space;

[0033] Under the second original model space, determine the second original agent sampling probability density function, and determine the second target agent importance sampling probability density function based on the second original agent sampling probability density function;

[0034] The second input variable sample is extracted from the first sample pool based on the second target agent importance sampling probability density function, and the second failure domain indicator function is calculated based on the second input variable sample and the adaptive Kriging agent model.

[0035] Calculate the difference between the first multidimensional distribution parameter variable and the second multidimensional distribution parameter variable to obtain the third multidimensional distribution parameter variable, and transform the third variable integration point of the third multidimensional distribution parameter variable to the second original model space to obtain the third original model space;

[0036] In the third original model space, based on the second input variable sample and the second subjective probability density function, the second β-sphere indicator function, the second failure domain indicator function, and the third variable integration point of the second input variable sample, the second failure probability integration point of the output failure probability of the real radome structure model is calculated.

[0037] The first conditional expected value of the output failure probability is calculated based on the second failure probability integral point, and the first conditional variance of the output failure probability is calculated based on the first conditional expected value and the third weight value.

[0038] In one exemplary embodiment of this disclosure, calculating the second conditional variance value of the output failure probability according to the adaptive Kriging surrogate model includes:

[0039] Calculate the fourth variable integration point and the fourth weight value of the third multidimensional distribution parameter variable, and transform the fourth variable integration point to the independent standard normal space to obtain the fourth original model space;

[0040] In the fourth original model space, the sampling probability density function of the third original agent is determined, and the sampling probability density function of the third target agent is determined based on the sampling probability density function of the third original agent.

[0041] The third input variable sample is extracted from the first sample pool based on the third target agent important sampling probability density function, and the third failure domain indicator function is calculated based on the second input variable sample and the adaptive Kriging agent model.

[0042] The second variable integral point of the second multidimensional distribution parameter variable is transformed to the independent standard normal space to obtain the fifth original model space. In the fifth original model space, the third failure probability integral point of the output failure probability is calculated based on the third input variable sample and the third subjective probability density function, the third β ball indicator function, the third failure domain indicator function of the third variable input sample and the second variable integral point.

[0043] The second conditional variance of the output failure probability is calculated based on the third failure probability integral point and the second weight value of the second multidimensional distribution parameter variable.

[0044] In one exemplary embodiment of this disclosure, calculating the conditional expected value of the output failure probability based on the second conditional variance value includes:

[0045] The conditional expected value of the output failure probability is calculated based on the second conditional variance value and the second weight value of the second multidimensional distribution parameter variable.

[0046] In one exemplary embodiment of this disclosure, the method for analyzing the reliability sensitivity of the distributed parameters of the radome structure further includes:

[0047] Obtain the subjective probability density function and the original failure domain indication function of the original distribution parameters of the real radome structure model, and calculate the output failure probability of the real radome structure model based on the subjective probability density function and the original failure domain indication function.

[0048] In one exemplary embodiment of this disclosure, the main sensitivity is used to measure the contribution of any original distribution parameter alone to the total variance of the output failure probability;

[0049] The total sensitivity is used to measure the contribution of any original distributed parameter, acting alone and interacting with other distributed parameters, to the total variance of the output failure probability.

[0050] This disclosure provides a method for analyzing the reliability sensitivity of distributed parameter uncertainty in a radome structure. On one hand, it constructs a first initial training sample set based on the original distributed parameters of a real radome structure model, and then constructs an adaptive Kriging surrogate model based on this first initial training sample set. Next, it calculates the total variance and first conditional variance of the output failure probability of the real radome structure model based on the adaptive Kriging surrogate model, and calculates the principal sensitivity of the real radome structure model based on the total variance and the first conditional variance. Finally, it calculates the second conditional variance of the output failure probability based on the adaptive Kriging surrogate model, and calculates the output failure probability based on the second conditional variance. The conditional expected value of the rate is calculated. Finally, the total sensitivity of the real radome structure model is calculated based on the second conditional variance and the conditional expected value. The sensitivity of the real radome structure model is analyzed based on the principal sensitivity and the total sensitivity. This realizes the sensitivity analysis of the real radome structure model based on the adaptive Kriging surrogate model, which solves the problem that the existing technology cannot model the radome structure and efficiently analyze the sensitivity of the radome structure based on this model, thus improving the efficiency of sensitivity analysis. On the other hand, since the sensitivity of the real radome structure model can be analyzed based on the principal sensitivity and the total sensitivity, the accuracy of the reliability sensitivity analysis results is improved.

[0051] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit this disclosure. Attached Figure Description

[0052] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure. It is obvious that the drawings described below are merely some embodiments of this disclosure, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0053] Figure 1 The flowchart illustrates a method for analyzing the reliability sensitivity of distributed parameter uncertainty of a radome structure according to an exemplary embodiment of the present disclosure.

[0054] Figure 2The schematic diagram illustrates a honeycomb sandwich radome structure model according to an exemplary embodiment of the present disclosure.

[0055] Figure 3 The flowchart illustrates a method for constructing a first initial training sample set based on the original distribution parameters of a real radome structure model, according to an example embodiment of the present disclosure.

[0056] Figure 4 The flowchart illustrates a method for constructing an adaptive Kriging proxy model based on a first initial training sample set according to an example embodiment of the present disclosure.

[0057] Figure 5 This illustration schematically shows an unconditional failure probability varying with N according to an example embodiment of the present disclosure. X Example graph of an increasing convergence curve.

[0058] Figure 6 An example diagram illustrating the comparison results of a primary sensitivity according to an exemplary embodiment of the present disclosure is shown.

[0059] Figure 7 An example diagram illustrating a comparison result of total sensitivity according to an exemplary embodiment of the present disclosure is shown.

[0060] Figure 8 The diagram schematically illustrates a block diagram of a distributed parameter uncertainty reliability sensitivity analysis apparatus for a radome structure according to an exemplary embodiment of the present disclosure.

[0061] Figure 9 An electronic device is illustrated according to an example embodiment of the present disclosure for implementing the distributed parameter uncertainty reliability sensitivity analysis method for the radome structure described above. Detailed Implementation

[0062] Example embodiments will now be described more fully with reference to the accompanying drawings. However, example embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided to make this disclosure more comprehensive and complete, and to fully convey the concept of the example embodiments to those skilled in the art. The described features, structures, or characteristics can be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a full understanding of embodiments of this disclosure. However, those skilled in the art will recognize that the technical solutions of this disclosure can be practiced with one or more of the specific details omitted, or other methods, components, apparatus, steps, etc., can be employed. In other instances, well-known technical solutions are not shown or described in detail to avoid obscuring various aspects of this disclosure.

[0063] Furthermore, the accompanying drawings are merely illustrative of this disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities may be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0064] This exemplary embodiment first provides a method for analyzing the reliability and sensitivity of distributed parameter uncertainties in a radome structure. This method can run on a server, server cluster, or cloud server, etc. Of course, those skilled in the art can also run the method disclosed herein on other platforms as needed, and this exemplary embodiment does not impose any special limitations on this. Specifically, refer to... Figure 1 As shown, the reliability and sensitivity analysis method for this radome structure may include the following steps:

[0065] Step S110. Construct a first initial training sample set based on the original distribution parameters of the real radome structure model, and construct an adaptive Kriging proxy model based on the first initial training sample set;

[0066] Step S120. Calculate the total variance and the first conditional variance of the output failure probability of the real radome structure model according to the adaptive Kriging surrogate model, and calculate the main sensitivity of the real radome structure model according to the total variance and the first conditional variance.

[0067] Step S130. Calculate the second conditional variance of the output failure probability according to the adaptive Kriging surrogate model, and calculate the conditional expectation of the output failure probability according to the second conditional variance.

[0068] Step S140. Calculate the total sensitivity of the real radome structure model based on the second conditional variance value and the conditional expectation value, and analyze the sensitivity of the real radome structure model based on the main sensitivity and the total sensitivity.

[0069] In the aforementioned reliability sensitivity analysis method for radome structures, on the one hand, a first initial training sample set can be constructed based on the original distribution parameters of the real radome structure model, and an adaptive Kriging surrogate model can be constructed based on the first initial training sample set; then, the total variance and the first conditional variance of the output failure probability of the real radome structure model are calculated based on the adaptive Kriging surrogate model, and the principal sensitivity of the real radome structure model is calculated based on the total variance and the first conditional variance; then, the second conditional variance of the output failure probability is calculated based on the adaptive Kriging surrogate model, and the conditional expectation of the output failure probability is calculated based on the second conditional variance. Finally, the total sensitivity of the real radome structure model is calculated based on the second conditional variance and conditional expectation. The sensitivity of the real radome structure model is then analyzed based on the principal sensitivity and the total sensitivity. This enables sensitivity analysis of the real radome structure model based on an adaptive Kriging surrogate model, solving the problem in existing technologies that cannot model radome structures and efficiently analyze their sensitivity based on this model, thus improving the efficiency of sensitivity analysis. On the other hand, since the sensitivity of the real radome structure model can be analyzed based on the principal sensitivity and the total sensitivity, the accuracy of the reliability sensitivity analysis results is further improved.

[0070] The following will provide a detailed explanation and description of the distributed parameter uncertainty, reliability, and sensitivity analysis method for the radome structure of the present disclosure, in conjunction with the accompanying drawings.

[0071] First, the application scenarios and inventive objectives of the exemplary embodiments of this disclosure will be explained and described. Specifically, the reliability sensitivity analysis method for the distributed parameter uncertainty of the radome structure described in the exemplary embodiments of this disclosure relates to the global reliability sensitivity analysis problem of the radome structure when the distributed parameters have uncertainty. Furthermore, the reliability sensitivity analysis method for the radome structure described in the exemplary embodiments of this disclosure provides an efficient global reliability sensitivity analysis algorithm to address the difficulties of traditional methods. The main techniques applied in this global reliability sensitivity analysis algorithm include: on the one hand, the Cubature Formula (CF); on the other hand, the Surrogate Sampling Probability Density Function (SSPDF); furthermore, the Kriging method; and, truncated Importance Sampling (TIS).

[0072] The reliability sensitivity analysis method for distributed parameter uncertainty of radome structures described in this exemplary embodiment aims to efficiently analyze the influence of each distributed parameter in the radome structure on the output failure probability, thereby improving its reliability and safety in practical applications. In the specific implementation process, a proxy model for establishing a real radome structure model can be established by introducing the SSPDF method (STISPDF method) combined with TIS into the adaptive Kriging proxy model method. Based on this proxy model, the CF method is further applied to the global reliability sensitivity analysis under distributed parameter uncertainty, proposing a method for efficiently solving the global reliability sensitivity index of radome structure distributed parameters: the quadrature formula (AKS-CF) method based on the adaptive Kriging proxy model combined with the TIS method and the SSPDF method.

[0073] Secondly, the actual radome structure model described in the exemplary embodiments of this disclosure will be explained and illustrated. Specifically, the example diagram of the actual radome structure model can be found in [reference needed]. Figure 2 As shown. Specifically, in Figure 2 The actual radome structure model shown can contain 15 intensity variables, and the distribution parameters of each intensity variable can be 21-dimensional variables; the specific details of each intensity variable and its corresponding distribution parameters are shown in Table 1 below:

[0074] Table 1 Distribution parameters of input variables for radome structure

[0075]

[0076]

[0077] Furthermore, it is assumed that the original input variables and their means of the real radome structure model have uncertainty and their distribution information is listed in Table 1. The input variables and their means follow normal and uniform distributions, respectively, and the coefficient of variation of each input variable is 0.2. At the same time, there is a constraint relationship between the original input variables and the original distribution parameters; that is, the original input variables can be constrained by the original distribution parameters. Therefore, considering the buckling failure analysis of this real radome structure model, the definition of its failure domain can be shown in the following formula (1):

[0078] F = {x: g(x) = γ(x) - Force × M ≤ 0}; Formula (1)

[0079] Wherein, γ(x) is the buckling response function; Force is the wind load. Due to the complexity of the aerial environment, the actual wind load fluctuates greatly and needs to be re-examined from the perspective of uncertainty. For example, it is assumed that the original load is 1 and that the original wind load follows a normal distribution: N(1, 1×cov); where cov is the coefficient of variation of the wind load. In practical applications, when cov = 0.1, the overall wind load distribution is a normal distribution N(1, 0.1). When the sampling value is 0.95, the wind load at all locations is multiplied by 0.95. Furthermore, M is the safety margin (M>1) taken by the actual radome structure model. M is set to prevent the consequences caused by factors such as sudden increases in wind load. Therefore, the test cannot continue to be carried out according to the given load. Instead, the original load value is multiplied by the safety margin M. This actually increases the safety test standard of the structure and puts forward more stringent requirements for the structural design, but also makes the structure more reliable. Furthermore, combined with the actual situation, the coefficient of variation of the wind load in this disclosure is cov = 0.2 and the safety margin is M = 3.

[0080] Next, the output failure probability of the real radome structure model described in the example embodiments of this disclosure will be explained and illustrated. Specifically, the calculation method for the output failure probability is as follows: obtain the subjective probability density function and the original failure domain indicator function of the original distribution parameters of the real radome structure model, and calculate the output failure probability of the real radome structure model based on the subjective probability density function and the original failure domain indicator function. That is, in practical applications, for structures with parameter uncertainties, its function is expressed as: Y = g(X,Θ); where, It is d X The input variables are mutually independent, which are the original distribution parameters described above; It is the d of the input variable X Θ There are 3 independent distributed parameter variables; where the subjective uncertainty of the parameters can be represented by the subjective probability density function f. Θ The output failure probability is described by (θ). In this case, the output failure probability is no longer solely influenced by the uncertainty of the input variable X, but can be described as a function of the distribution parameter Θ, i.e.:

[0081]

[0082] Among them, f X (x|θ) is the subjective probability density function of the input variable X given the distribution parameters. For the space of input variables, I F (x|θ) is the failure domain indicator function, when x∈I F When (x|θ), then I F (x|θ)=1; otherwise, I F (x|θ)=0;Pf The true value is the output failure probability of the radome structure model. Therefore, similar to the variance-based sensitivity analysis method, the global reliability sensitivity index and the total sensitivity index of the distributed parameters to the output failure probability can be defined as follows: Equation (3) and Equation (4):

[0083]

[0084]

[0085] Among them, the principal sensitivity measures the contribution of any single original distribution parameter to the total variance of the output failure probability; the total sensitivity measures the contribution of any single original distribution parameter, as well as its interaction with other distribution parameters, to the total variance of the output failure probability; that is, the principal sensitivity index S i Used to measure the distribution parameter Θ i The failure probability P under single action f The contribution of variance, the total sensitivity index is used to measure the distribution parameter Θ. i The effect of individual action and interaction with other distribution parameters on the failure probability P f The contribution of variance.

[0086] Next, the usage principle of the quadrature formula involved in the example embodiments of this disclosure will be explained and described. Specifically, it can be seen from the above formulas (3) and (4) that the key to solving the index lies in the calculation of nested expectation and variance; therefore, in order to efficiently solve the nested expectation and variance operators, this disclosure uses the theory of CF (quadrature formula) to simplify this solution process; wherein, the statistical moments of the CF estimated variables are performed in the standard normal space, therefore, the model of formula (2) is transformed into independent standard normal variables through Rosenblatt transformation or Nataf transformation. The function can be specifically represented by the following formula (5):

[0087] P f =ψ(Θ)=ψ[R -1 [(λ)]=ρ(λ); Formula (5)

[0088] Among them, R -1 (·) represents the Rosenblatt transform or Nataf transform, and ρ(λ) represents a multivariate function of the independent standard normal variable λ; at the same time, according to formula (5), the output failure probability P f The expected value and variance can be expressed in integral form as shown in the following formulas (6) and (7), respectively:

[0089]

[0090]

[0091] Among them, f λ (λ) represents the joint probability density function of the independent standard normal variables λ. Furthermore, according to the theory of quadrature formulas, the integrals in formulas (6) and (7) can be efficiently calculated using a small number of suitable integration points and corresponding weights, as shown in formulas (8) and (9) below:

[0092]

[0093]

[0094] Based on the foregoing content, it can be seen that the quadrature formula used in this disclosure to calculate the nested expectation and variance can be specifically expressed as formula (10) below:

[0095]

[0096] Where I() denotes the abbreviation of the integral, if we consider the integral in formula (8) and in formula (9) Then r() in formula (10) can be expressed as ρ(λ) and

[0097] Furthermore, the principle of the proxy truncated important sampling probability density function involved in the example embodiments of this disclosure is explained and illustrated. Specifically, by introducing CF, nested expectations and variances can be solved efficiently and accurately. However, when the uncertainty of the distribution parameters is transferred to the output failure probability, two-level sampling of the input variables and distribution parameters is required. To solve this problem, this disclosure uses the SSPDF method. In practical applications, this method solves the problem that the computational load depends on the parameter dimension in the process of uncertainty transfer from distribution parameters to failure probability, and has the advantages of reducing the number of nested sampling layers and improving the efficiency of uncertainty transfer. For example, by introducing SSPDF into the index solution, the output failure probability in formula (2) can be shown in formula (11) below.

[0098]

[0099] Among them, h X (x|θ * ) is SSPDF, This represents the average value of SSPDF; meanwhile, according to formula (11), it can be seen that when the calculation process of the output failure probability is introduced into SSPDFh... X (x|θ * This is generated independently of the true distribution parameters. Therefore, in calculating P... fWhen the distribution parameter Θ changes in the outer layer, the samples of the input variable X generated in the inner layer can be reused.

[0100] The above analysis shows that SSPDFh X (x|θ * The selection of h is crucial. In some technical solutions, when the input variable X changes with its uncertain distribution parameter Θ, h... X (x|θ * The response should cover the entire range of variation of the input variable X. One simple and direct approach is to determine the limiting distribution of the input variable X based on the range of variation of the distribution parameter Θ, and then determine SSPDFh based on that limiting distribution. X (x|θ * To effectively improve the sampling efficiency of radome structures under low failure probabilities, this disclosure introduces the concept of the TIS method into the SSPDF method, thereby constructing the STISPDF. The TIS method described here combines the concepts of importance sampling and truncated sampling. The basic idea of ​​importance sampling is to extract samples through an importance sampling probability density function, ensuring that a large number of the extracted sample points fall within the failure domain. This allows the estimated failure probability to converge to the true value more quickly, effectively solving the problem of low failure probabilities. The point in the failure domain of the function that contributes the most to the failure probability is the design point; therefore, the density center is generally set at the design point when constructing the importance sampling density function. This invention further shifts the sampling center of SSPDF to the design point to construct an importance sampling surrogate density function (SISPDF) to further improve efficiency. The basic idea of ​​the truncated importance sampling method is to establish a β hypersphere in the standard normal space with the origin as the center and the reliability index (the shortest distance from the origin to the limit state surface) as the radius. When sample points are extracted using SISPDF, the sample points falling into the β hypersphere are within the safe region. Therefore, this part of the sample points can be truncated, thereby reducing the amount of calculation when estimating the failure probability.

[0101] Furthermore, the indicator function within the region outside the β hypersphere can be defined as I. β (x). Wherein, the indicator function I β (x) can be specifically represented by the following formula (12):

[0102]

[0103] Therefore, the expression for the failure probability at this time can be shown by the following formula (13):

[0104]

[0105] in, Indicates SISPDF, E * [] indicates the average value for SISPDF.

[0106] The following section explains and illustrates the principles behind using the Kriging surrogate model combined with truncated importance sampling and the surrogate sampling probability density function method. Specifically, for low failure probability problems in engineering structures, calculating the structural output requires repeated calls to the function, especially for complex implicit structural systems. The cost of repeatedly calling the finite element model is extremely high. Therefore, this invention combines truncated importance sampling and the surrogate sampling probability density function to first establish a surrogate model for the radome structure.

[0107] Since the original Kriging proxy model cannot resolve the trade-off between accuracy and efficiency, an adaptive proxy model is needed to address this issue; and, The learning function considers the distance between the Kriging surrogate model's predicted value and the failure surface, as well as the standard deviation of the estimated value. After establishing a coarse Kriging model using a small number of sample points, this learning function selects qualified sample points from the remaining candidate sample points and adds them to the current training sample set to update the Kriging model. It selects minU(x)≥2 as the convergence termination condition for the adaptive update process of the Kriging surrogate model. Furthermore, the U learning function is only applicable to structural systems with a limit state surface of 0. For a system with a threshold of e and failure when Y=g(X)≤e, a new function Y needs to be constructed based on the original function and the failure condition. reb =g reb (X) = g(X) - e, and then the U learning function is used to establish a description of Y. reb =g reb The Kriging model of (X). Therefore, in the reliability and sensitivity analysis method of the radome structure described in the example embodiments of this disclosure, an AK-STISPDF surrogate model will also be established based on the U learning function; that is, the AK-STISPDF method is used to establish a surrogate model of the radome structure input-output, and based on this surrogate model, the global reliability and sensitivity index of the distributed parameters is calculated using CF. At the same time, the AKS-CF method can also be established, which solves the parameter Θ i and Θ i When calculating the output failure probability under certain conditions, formula (13) can be transformed to obtain the following formulas (14) and (15):

[0108]

[0109]

[0110] Where, θ i ′ and θ′ iThey respectively represent the parameters Θ i and Θ i Under the condition of fixed parameter values. Therefore, the conditional output failure probability of the index recorded in formula (3) and formula (4) can be calculated by formula (14) and formula (15) respectively, and then the index S can be calculated. i as well as This allows for the analysis of the sensitivity of a real radome structural model.

[0111] The following will combine the above principles to... Figure 1 The reliability and sensitivity analysis method for the radome structure shown will be further explained and illustrated. Specifically:

[0112] In step S110, a first initial training sample set is constructed based on the original distribution parameters of the real radome structure model, and an adaptive Kriging surrogate model is constructed based on the first initial training sample set.

[0113] In this example embodiment, firstly, a first initial training sample set is constructed based on the original distribution parameters of the actual radome structure model. Specifically, refer to... Figure 3 As shown, the following steps may be included:

[0114] Step S310: Obtain the original distribution parameters of the real radome structure model, and use the first target proxy importance sampling probability density function to sample the original input variables to obtain multiple sampling results;

[0115] Step S320: Construct a first sample pool based on the multiple sampling results, and randomly extract multiple first original sample points from the first sample pool;

[0116] Step S330: Calculate the first distance between the first original sample point and the origin of the real radome structure model, and select multiple first target sample points from the first original sample points according to the first distance;

[0117] Step S340: Calculate the first buckling response value of each of the first target sample points using the original model function of the real radome structure model, and construct the first initial training sample set based on the first buckling response value.

[0118] In one example embodiment, sampling the original input variables using a first target surrogate importance sampling probability density function to obtain multiple sampling results can be achieved as follows: First, based on the selection principle of the surrogate sampling probability density function, determine the first original surrogate sampling probability density function of the real radome structure model; second, in the independent standard normal space, calculate the design point and reliability index of the real radome structure model using the improved first-order second-moment method; then, shift the first original surrogate sampling probability density function from the sampling center to the design point to obtain the first target surrogate importance sampling probability density function; finally, use the first target surrogate importance sampling probability density function to sample the original input variables to obtain multiple sampling results.

[0119] The following will explain and illustrate steps S310-S340. Specifically, in practical applications, firstly, the first original surrogate sampling probability density function h of the radome structure model can be determined according to the selection principle of SSPDF (surrogate sampling probability density function). X (x|θ * Then, in the independent standard normal space, the design points and reliability index β of the real radome structure model are solved using the AFOSM (Advanced First Order and Second Moment) method. Finally, the first original surrogate sampling probability density function h is used. X (x|θ * The sampling center is shifted to the design point to construct the first target agent importance sampling probability density function SISPDF. Furthermore, the first target proxy importance sampling probability density function SISPDF is used. Extract N from the input variable (original distribution parameters) X K Sample (sampling results) x k (k=1,,N K ), and form the first sample pool S based on the samples. TIS Furthermore, from the first sample pool S TIS N are randomly selected from the middle T (N T N K Given 10 input variable samples (the first original sample points), calculate the first original sample point. Distance to the origin And then select those that meet the requirements The first target sample points form the new input variable sample x k (k=1,,N′ T Finally, the buckling response value of each first target sample point is calculated, thereby constructing the first initial training sample set T.TIS ={(x k ,γ(x k )-Force×M),k=1,,N′ T}

[0120] Secondly, after obtaining the first initial training sample set, an adaptive Kriging proxy model can be constructed based on it. For details, refer to... Figure 4 As shown, the following steps may be included:

[0121] Step S410: Construct a target function based on the original model function of the real radome structure model and the preset failure conditions, and construct the initial Kriging proxy model based on the target function and the first initial training sample set.

[0122] Step S420: Calculate the U learning function value of the sampling results included in the first sample pool using the initial Kriging proxy model, and select sample points to be updated from the sampling results in the first sample pool according to the U learning function value.

[0123] Step S430: When it is determined that the U learning function value of the sample point to be updated is greater than or equal to a preset threshold, the initial Kriging proxy model is used as the adaptive Kriging proxy model.

[0124] The following will explain and illustrate steps S410-S430. Specifically, firstly, based on the first initial training sample set T... TIS Use the DACE (Design and Analysis of Computer Experiments) toolbox to establish the function Y reb =γ(X) - Force×M and the first initial Kriging surrogate model g of X eK (X); Then, the first sample pool S is calculated using the first initial kriging model. TIS For each sample point (sampling result), calculate the U learning function value and select the next sample point to be updated (the sample point to be updated). when If the adaptive learning process is stopped at a certain point, the adaptive Kriging surrogate model has been established, meaning the initial Kriging surrogate model can be used as the adaptive Kriging surrogate model; of course, if Then it is necessary to make {x u ,g(x u )=γ(x u )-Force×M} is added to the first initial training sample set T TISThe initial Kriging surrogate model is then reconstructed based on the updated first initial training sample set until the U-learning function value of the sample points to be updated is greater than or equal to 2. It should be noted that during the adaptive learning process of the first initial Kriging model, the target training sample set can be obtained by continuously updating the first initial training sample set using sample points with U-learning function values ​​less than 2. The initial Kriging surrogate model is then reconstructed based on the updated target training sample set, thus achieving adaptive learning of the initial Kriging surrogate model to obtain an adaptive Kriging surrogate model. This improves the accuracy of the obtained output failure probability and ultimately enhances the accuracy of the reliability sensitivity analysis results.

[0125] In step S120, the total variance and the first conditional variance of the output failure probability of the real radome structure model are calculated according to the adaptive Kriging surrogate model, and the main sensitivity of the real radome structure model is calculated based on the total variance and the first conditional variance.

[0126] In this example embodiment, firstly, the total variance of the output failure probability of the real radome structure model is calculated based on the adaptive Kriging surrogate model. Specifically, this can be achieved as follows: First, the first multidimensional distribution parameter variable of the original distribution parameter is calculated in an independent standard normal space, and the first variable integral point and the first weight value of the first multidimensional distribution parameter variable are calculated; secondly, the independent standard normal space is transformed using the first variable integral point to obtain a first original model space, and multiple sampling results are selected from the first sample pool to construct a first input variable sample in the first original model space; then, the first failure domain indicator function is calculated based on the adaptive Kriging surrogate model and the first input variable sample, and the first failure probability integral point of the output failure probability of the real radome structure model is calculated based on the first target surrogate importance sampling probability density function, the first subjective probability density function, the first β-sphere indicator function, the first failure domain indicator function, and the first variable integral point of the first input variable sample; finally, the total variance of the output failure probability is calculated based on the first failure probability integral point and the first weight value.

[0127] The following will explain the specific calculation process of the total variance of the output failure probability. Specifically, in the independent standard normal space, the first variable integration point and the first weight value are calculated based on the formula (10) described above. Specifically, formula (10) can be as follows:

[0128]

[0129] Among them, the first multidimensional distribution parameter variable d is based on formula (10). ΘThe first variable integration point obtained from the calculation is The first weight value is Then, the integration points of the first variable are transformed to independent standard normal space to obtain the first original model space. Furthermore, when At that time, from the first sample pool S TIS N is selected from the input variable X. X Sample x k (k=1,,N X ), calculate each sample Distance to the origin And select those that meet the requirements The sample points form a new input variable sample x k (k=1,…,N′ X (First input variable sample); that is, I β (x k |θ * (k=1,…,N′) X =1, and after substituting the first input variable sample into the adaptive Kriging surrogate model, the first failure domain indicator function I is obtained. F (x k |θ * (k=1,…,N′) X Furthermore, based on the first target surrogate importance sampling probability density function of the first input variable sample... First subjective probability density function f X (x k |θ)(k=1,…,N′ X ), First β-ball indicator function I β (x k |θ * (k=1,…,N′) X and the first failure domain indication function I F (x k |θ * (k=1,…,N′) X ) and the first variable integration point of Θ The output failure probability P is calculated using formula (13). f First failure probability integration point Formula (13) can be expressed as follows:

[0130]

[0131] Finally, the integral point of the first failure probability can be used as a basis. and the first weight value P can be obtained by solving the aforementioned formula (10). fTotal variance V(P) f Formula (10) can be expressed as follows:

[0132]

[0133] Secondly, the first conditional variance of the output failure probability of the real radome structure model is calculated based on the adaptive Kriging surrogate model. Specifically, this can be achieved as follows: First, in the first original model space, the second variable integration point and the second weight value of the second multidimensional distribution parameter variable included in the first original model space are calculated, and the second variable integration point is transformed to the first original model space to obtain the second original model space; second, the second original surrogate sampling probability density function is determined in the second original model space, and the second target surrogate importance sampling probability density function is determined based on the second original surrogate sampling probability density function; then, based on the second target surrogate importance sampling probability density function, the second input variable sample is extracted from the first sample pool, and the second failure domain indicator function is calculated based on the second input variable sample and the adaptive Kriging surrogate model; further, the first multidimensional distribution is calculated. The difference between the parameter variable and the second multidimensional distribution parameter variable is used to obtain the third multidimensional distribution parameter variable. The third variable integration point of the third multidimensional distribution parameter variable is then transformed to the second original model space to obtain the third original model space. Furthermore, in the third original model space, based on the second input variable sample and the second subjective probability density function, the second β-sphere indicator function, the second failure domain indicator function, and the third variable integration point of the second input variable sample, the second failure probability integration point of the output failure probability of the real radome structure model is calculated. Finally, based on the second failure probability integration point, the first conditional expectation value of the output failure probability is calculated, and based on the first conditional expectation value and the third weight value, the first conditional variance value of the output failure probability is calculated.

[0134] The following will explain and illustrate the specific calculation process of the first conditional variance. Specifically, firstly, in the first original model space, the integral points of the second multidimensional distribution parameter variable (l-dimensional) are calculated according to the aforementioned quadrature formula (i.e., formula (10)). and the second weight value The second variable integration point Transform to the first original model space to obtain the second original model space. Secondly, when At that time, determine the new SSPDF (That is, the second original surrogate sampling probability density function), and construct SSPDF based on the second original surrogate sampling probability density function. (Second target agent importance sampling probability density function), and by the second target agent importance sampling probability density function Extract N of input variable X X Sample x′ k (k=1,,N X ), calculate each sample Distance to the origin Select the matching The sample points form a new input variable sample x′ k (k=1,,N′ X (Second input variable sample), that is Where (k=1,,N′) X The second input variable sample is then substituted into the adaptive Kriging surrogate model to obtain the second failure domain indicator function. Then, based on the quadrature formula described above (i.e., Formula 10), d is generated. Θ -1 dimension distribution parameter (third multidimensional parameter distribution variable) third variable integration point and the third weight value The integration point of the third variable Transforming to the second original model space yields the third original model space. Furthermore, when At that time, input samples according to the second variable. The second subjective probability density function of the second input variable sample Second β-ball indicator function Second Failure Domain Indication Function and Θ i The integration point (the integration point of the third variable) The output failure probability P under the aforementioned parameter conditions is calculated using formula (14). f Second failure probability integration point Furthermore, based on the second failure probability integration point and the third weight value... The output failure probability P is calculated using the formula (10) described above. f First conditional expectation Finally, based on the expected value of the first condition. and the third weight value The first conditional expected value of the output failure probability is calculated using the formula (10) described above.

[0135] At this point, the calculation process for the conditional expected value of the output failure probability and the first conditional expected value has been completed. Under this premise, the conditional expected value V(P) can then be used to calculate the expected value. f and the first conditional expected value Based on the aforementioned formula (3), the main sensitivity S of the real radome structure model is calculated. i .

[0136] In step S130, the second conditional variance of the output failure probability is calculated according to the adaptive Kriging surrogate model, and the conditional expected value of the output failure probability is calculated according to the second conditional variance.

[0137] In this example embodiment, firstly, the second conditional variance of the output failure probability is calculated based on the adaptive Kriging surrogate model. Specifically, this can be achieved as follows: First, calculate the fourth variable integral point and the fourth weight value of the third multidimensional distribution parameter variable, and transform the fourth variable integral point to an independent standard normal space to obtain the fourth original model space; second, in the fourth original model space, determine the third original surrogate sampling probability density function, and determine the third target surrogate importance sampling probability density function based on the third original surrogate sampling probability density function; then, based on the third target surrogate importance sampling probability density function, extract the third input variable sample from the first sample pool, and calculate the third failure domain indicator function based on the second input variable sample and the adaptive Kriging surrogate model; further, transform the second variable integral point of the second multidimensional distribution parameter variable to an independent standard normal space to obtain the fifth original model space, and in the fifth original model space, calculate the third failure probability integral point of the output failure probability based on the third input variable sample, the third subjective probability density function of the third variable input sample, the third β-ball indicator function, the third failure domain indicator function, and the second variable integral point; finally, calculate the second conditional variance value of the output failure probability based on the third failure probability integral point and the second weight value of the second multidimensional distribution parameter variable.

[0138] The following will explain and illustrate the specific calculation process of the second conditional variance. Specifically, firstly, the third multidimensional distribution parameter variable d is calculated according to the aforementioned quadrature formula (i.e., formula (10)). Θ -1, fourth variable integration point and the fourth weight value and the fourth variable integration point Transform to the independent standard normal space to obtain the fourth original model space Secondly At that time, determine the third original agent sampling probability density function SSPDF. And based on the sampling probability density function of the third original agent, the importance sampling probability density function SSPDF of the third target agent is constructed. And the important sampling probability density function is represented by the third target. N of the input variable X X Sample x′k (k=1,…,N) X ), calculate each sample Distance to the origin Select the matching The sample points form a new third input variable sample x′ k ′(k=1,…,N′ X ),Right now The third failure domain indication function is obtained by substituting the sample of the third input variable into the adaptive Kriging surrogate model. Then, the second variable integration point of the 1-dimensional distribution parameter (the second multidimensional distribution parameter variable) is generated according to the aforementioned quadrature formula (i.e., formula (10)). Second weight value The integration point of the second variable is obtained by using the Rosenblatt transform or the Nataf transform. Transforming to the independent standard normal space yields the fifth original model space. Furthermore, when At that time, based on the third input variable sample The third subjective probability density function of the third variable input sample Third β-ball indicator function Third Failure Domain Indication Function and Θ i Integral points (Second variable integration point), based on the aforementioned formula (15), solve for the output failure probability P. f The third failure probability integration point Then, from the third failure probability integration point and the second weight value The output failure probability P is calculated using the formula (10) described above. f Second conditional variance

[0139] At this point, the specific calculation process for the second conditional variance has been fully implemented. Based on this, the conditional expected value of the output failure probability can be calculated using the second conditional variance value. Specifically, this can be achieved as follows: The conditional expected value of the output failure probability is calculated based on the second conditional variance value and the second weight value of the second multidimensional distribution parameter variable. That is, the conditional expected value of the output failure probability can be calculated from the second conditional variance value. and the second weight value The conditional expectation value is solved using the formula (10) described above.

[0140] In step S140, the total sensitivity of the real radome structure model is calculated based on the second conditional variance value and the conditional expectation value, and the sensitivity of the real radome structure model is analyzed based on the main sensitivity and the total sensitivity.

[0141] Specifically, after obtaining the second conditional variance and the conditional expectation, the second conditional variance V(P) can be... f and conditional expected value Substituting into the formula (4) described above, the total sensitivity can be obtained. Finally, the sensitivity of the real radome structure model is analyzed based on the main sensitivity and the total sensitivity.

[0142] In practical applications, the computational cost of repeatedly calling the original radome structure model when calculating the reliability sensitivity index using MCS is too high, making the analysis process difficult to implement. Therefore, this disclosure first compares the unconditional output failure probabilities calculated using the real radome structure model and the Kriging proxy model. An example of the comparison results is shown in Table 2 below. Specifically, in Table 2, N... X The sample size represents the number of input variables selected in the sample pool. NPFE represents the number of times the original model's performance function evaluation (NPFE) is called. The NPFE for establishing the Kriging surrogate model for the radome structure model is 206, while the NPFE for iteratively solving the design points using the AFOSM method is 521. Table 2 shows that the failure probability calculated by the Kriging surrogate model established using the AK-STISPDF method agrees well with the failure probability obtained by MCS using the original model, but the former's NPFE is much lower than MCS's. This indicates that the established Kriging surrogate model can reduce computational costs while maintaining computational accuracy, providing an analytical basis for radome models with complex structures.

[0143] Table 2 Unconditional Failure Probability of Radome Structure

[0144]

[0145] In addition, to visually demonstrate the superiority of the Kriging model established using the AK-STISPDF method, Figure 5 The table lists the failure probabilities obtained using the MCS method as a comparison solution, and the failure probabilities solved using the Kriging surrogate model as a function of N. X The convergence curve during the change. From Figure 5 It can be found that in N XWhen the value is 5000, the failure probability can converge and meet the calculation accuracy requirements, which shows that the Kriging model established by the AK-STISPDF method can replace the original model to realize the reliability sensitivity analysis of the structure.

[0146] Furthermore, to compare the efficiency of the AKS-CF method, this application replaces the implicit function with the established Kriging surrogate model and uses the Monte Carlo method based on the Kriging model (denoted as AKS-MCS) as the comparison method. The results obtained by the two methods are listed in Table 3. The values ​​in parentheses in the table are the coefficients of variation of the corresponding results, which are obtained by iteratively calculating the reliability sensitivity index and failure probability 30 times. N is the number of times the Kriging surrogate model is called. The N values ​​for calculating the reliability sensitivity index by the AKS-MCS and AKS-CF methods are respectively d Θ ×N Θ ×N Θ ×N X +N Θ ×N X as well as

[0147] Table 3 Reliability and Sensitivity Indicators of Radome Structure Distribution Parameters

[0148]

[0149] It should be noted that, due to the large number of structural variables, only numerical results with indices greater than 0.01 are listed in the table. The results in the table show that, provided the calculation results converge, the AKS-CF method yields good computational accuracy. The N value for obtaining the convergent solution using this method is 8.4425 × 10⁻⁶. 7 Compared to the MCS method's 8.4004×10 10 This significantly reduces the computational load, demonstrating high computational efficiency. Furthermore, to clearly compare the magnitudes of each indicator, Figure 6 as well as Figure 7 The image shows bar charts of the indices obtained from the two methods. From these, we can summarize the importance ranking of the distribution parameters obtained from the main and total sensitivity indices as μ. M2 >μ Mat3G13 >μ M1 >μ Mat1E22 >μ M3 >μ Mat1G12 This indicates that the parameter μ of the honeycomb interlayer thickness... M2 The individual effect of μ and its interaction with other parameters have the greatest impact on the failure probability; therefore, the parameter μ... M2 Effective adjustment can minimize the probability of structural failure. Additionally, the parameter μ... Mat3G13 μM2 and μ Mat1E22 The distribution parameters have a significant impact on the probability of structural failure, requiring extensive collection of subjective information to improve structural reliability. Other distribution parameters, which have almost no effect on the failure probability, can be fixed to arbitrary values ​​within the uncertainty range to simplify the structural analysis and design process.

[0150] Thus, the distributed parameter uncertainty reliability sensitivity analysis method for radome structures described in the exemplary embodiments of this disclosure has been fully implemented. Based on the foregoing description, it can be understood that the distributed parameter uncertainty reliability sensitivity analysis method for radome structures described in the exemplary embodiments of this disclosure addresses the problem that conventional methods struggle to achieve global reliability sensitivity analysis of high-dimensional nonlinear radome structures. This disclosure combines the SSPDF method and moves the sampling center of SSPDF to the design point, ensuring that more sample points fall within the failure domain. Using an established β-sphere, sample points falling within the safe domain are truncated, constructing the STISPDF method. An AK-STISPDF surrogate model for the radome structure is then established based on STISPDF, thereby avoiding repeated calls to the function when calculating the global reliability sensitivity index of the distributed parameters. This invention overcomes, to some extent, the problem of massive computational load in the index solution process. Furthermore, it applies the traditional quadrature formula to structural models with subjective parameter uncertainties, efficiently solving the nested expectation and variance in the global reliability sensitivity index of distributed parameters. This further establishes the AKS-CF method for efficiently solving the parameter reliability sensitivity index of radome structures. Moreover, this invention uses the proposed new algorithm to analyze radome structures, proving the effectiveness of the AK-STISPDF method in establishing the Kriging surrogate model. The proposed method can improve the computational efficiency of reliability sensitivity analysis while ensuring the accuracy of index and failure probability calculations, demonstrating good applicability. Additionally, the importance ranking of parameters is derived from the calculation results, which can provide guidance for reducing the failure probability of structures and simplifying the analysis and design process.

[0151] This disclosure also provides an example embodiment of a device for analyzing the reliability and sensitivity of distributed parameter uncertainties in a radome structure. Specifically, refer to... Figure 8 As shown, the reliability sensitivity analysis device for this radome structure may include an adaptive Kriging surrogate model construction module 810, a main sensitivity calculation module 820, a conditional expectation value calculation module 830, and a reliability sensitivity analysis module 840. Wherein:

[0152] The adaptive kriging surrogate model building module 810 can be used to build a first initial training sample set based on the original distribution parameters of the real radome structure model, and to build an adaptive kriging surrogate model based on the first initial training sample set.

[0153] The main sensitivity calculation module 820 can be used to calculate the total variance and the first conditional variance of the output failure probability of the real radome structure model according to the adaptive Kriging surrogate model, and to calculate the main sensitivity of the real radome structure model according to the total variance and the first conditional variance.

[0154] The conditional expected value calculation module 830 can be used to calculate the second conditional variance value of the output failure probability according to the adaptive Kriging surrogate model, and calculate the conditional expected value of the output failure probability according to the second conditional variance value.

[0155] The reliability sensitivity analysis module 840 can be used to calculate the total sensitivity of the real radome structure model based on the second conditional variance value and the conditional expectation value, and to analyze the sensitivity of the real radome structure model based on the main sensitivity and the total sensitivity.

[0156] In one exemplary embodiment of this disclosure, constructing a first initial training sample set based on the original distribution parameters of a real radome structure model includes: obtaining the original distribution parameters of the real radome structure model, and sampling the original input variables using a first target surrogate importance sampling probability density function to obtain multiple sampling results; constructing a first sample pool based on the multiple sampling results, and randomly selecting multiple first original sample points from the first sample pool; calculating a first distance between the first original sample points and the origin of the real radome structure model, and selecting multiple first target sample points from the first original sample points based on the first distance; calculating a first buckling response value for each first target sample point using the original model function of the real radome structure model, and constructing the first initial training sample set based on the first buckling response value.

[0157] In one exemplary embodiment of this disclosure, the original input variables are sampled using a first target surrogate importance sampling probability density function to obtain multiple sampling results. This includes: determining a first original surrogate sampling probability density function for the real radome structure model based on the selection principle of the surrogate sampling probability density function; calculating the design point and reliability index of the real radome structure model in an independent standard normal space using an improved first-order second-moment method; shifting the first original surrogate sampling probability density function from the sampling center to the design point to obtain a first target surrogate importance sampling probability density function; and sampling the original input variables using the first target surrogate importance sampling probability density function to obtain multiple sampling results.

[0158] In one exemplary embodiment of this disclosure, constructing an adaptive Kriging surrogate model based on a first initial training sample set includes: constructing a target function function based on the original model function of the real radome structure model and preset failure conditions; constructing the initial Kriging surrogate model based on the target function function and the first initial training sample set; calculating the U-learning function value of the sampling results included in the first sample pool using the initial Kriging surrogate model; selecting sample points to be updated from the sampling results in the first sample pool based on the U-learning function value; and using the initial Kriging surrogate model as an adaptive Kriging surrogate model when the U-learning function value of the sample point to be updated is greater than or equal to a preset threshold.

[0159] In one exemplary embodiment of this disclosure, calculating the total variance of the output failure probability of the real radome structure model based on an adaptive Kriging surrogate model includes: calculating a first multidimensional distribution parameter variable of the original distribution parameters in an independent standard normal space, and calculating a first variable integration point and a first weight value of the first multidimensional distribution parameter variable; transforming the independent standard normal space using the first variable integration point to obtain a first original model space, and constructing a first input variable sample by selecting multiple sampling results from a first sample pool in the first original model space; calculating a first failure domain indicator function based on the adaptive Kriging surrogate model and the first input variable sample, and calculating a first failure probability integration point of the output failure probability of the real radome structure model based on the first target surrogate importance sampling probability density function, the first subjective probability density function, the first β-sphere indicator function, the first failure domain indicator function, and the first variable integration point of the first input variable sample; and calculating the total variance of the output failure probability based on the first failure probability integration point and the first weight value.

[0160] In one exemplary embodiment of this disclosure, calculating the first conditional variance of the output failure probability of the real radome structure model based on the adaptive Kriging surrogate model includes: calculating the second variable integration point and second weight value of the second multidimensional distribution parameter variable included in the first original model space in the first original model space, and transforming the second variable integration point to the first original model space to obtain the second original model space; determining the second original surrogate sampling probability density function in the second original model space, and determining the second target surrogate importance sampling probability density function based on the second original surrogate sampling probability density function; extracting second input variable samples from the first sample pool based on the second target surrogate importance sampling probability density function, and calculating the first conditional variance of the output failure probability of the real radome structure model based on the second input variable samples and the adaptive Kriging surrogate model. The system employs two failure domain indicator functions. It calculates the difference between the first and second multidimensional distribution parameter variables to obtain a third multidimensional distribution parameter variable, and transforms the third variable integration point of the third multidimensional distribution parameter variable to the second original model space to obtain the third original model space. In the third original model space, based on the second input variable sample and its second subjective probability density function, second β-sphere indicator function, second failure domain indicator function, and third variable integration point, it calculates the second failure probability integration point of the output failure probability of the real radome structure model. Based on the second failure probability integration point, it calculates the first conditional expected value of the output failure probability, and based on the first conditional expected value and the third weight value, it calculates the first conditional variance of the output failure probability.

[0161] In one exemplary embodiment of this disclosure, calculating the second conditional variance value of the output failure probability according to the adaptive Kriging surrogate model includes: calculating the fourth variable integration point and the fourth weight value of the third multidimensional distribution parameter variable, and transforming the fourth variable integration point to an independent standard normal space to obtain a fourth original model space; determining the third original surrogate sampling probability density function in the fourth original model space, and determining the third target surrogate importance sampling probability density function based on the third original surrogate sampling probability density function; drawing samples of the third input variable from the first sample pool based on the third target surrogate importance sampling probability density function, and according to... The second input variable sample and the adaptive Kriging surrogate model are used to calculate the third failure domain indicator function; the second variable integration point of the second multidimensional distribution parameter variable is transformed to an independent standard normal space to obtain the fifth original model space; and in the fifth original model space, the third failure probability integration point of the output failure probability is calculated based on the third input variable sample, the third subjective probability density function of the third variable input sample, the third β-ball indicator function, the third failure domain indicator function, and the second variable integration point; and the second conditional variance value of the output failure probability is calculated based on the third failure probability integration point and the second weight value of the second multidimensional distribution parameter variable.

[0162] In one exemplary embodiment of this disclosure, calculating the conditional expected value of the output failure probability based on the second conditional variance value includes: calculating the conditional expected value of the output failure probability based on the second conditional variance value and the second weight value of the second multidimensional distribution parameter variable.

[0163] In one exemplary embodiment of this disclosure, the distributed parameter uncertainty reliability sensitivity analysis device for the radome structure further includes:

[0164] The output failure probability calculation module can be used to obtain the subjective probability density function and the original failure domain indication function of the original distribution parameters of the real radome structure model, and calculate the output failure probability of the real radome structure model based on the subjective probability density function and the original failure domain indication function.

[0165] In one exemplary embodiment of this disclosure, the principal sensitivity is used to measure the contribution of any original distributed parameter alone to the total variance of the output failure probability; the total sensitivity is used to measure the contribution of any original distributed parameter alone and its interaction with other distributed parameters to the total variance of the output failure probability.

[0166] The specific details of each module in the reliability and sensitivity analysis device for the aforementioned radome structure have been described in detail in the corresponding sensitivity method for the radome structure, so they will not be repeated here.

[0167] It should be noted that although several modules or units of the device for performing actions have been mentioned in the detailed description above, this division is not mandatory. In fact, according to embodiments of this disclosure, the features and functions of two or more modules or units described above can be embodied in one module or unit. Conversely, the features and functions of one module or unit described above can be further divided and embodied by multiple modules or units. Furthermore, although the steps of the method in this disclosure are described in a specific order in the accompanying drawings, this does not require or imply that these steps must be performed in that specific order, or that all the steps shown must be performed to achieve the desired result. Additional or alternative steps may be omitted, multiple steps may be combined into one step, and / or one step may be broken down into multiple steps.

[0168] In an exemplary embodiment of this disclosure, an electronic device capable of implementing the above-described method is also provided.

[0169] Those skilled in the art will understand that various aspects of this disclosure can be implemented as a system, method, or program product. Therefore, various aspects of this disclosure can be specifically implemented in the following forms: a completely hardware implementation, a completely software implementation (including firmware, microcode, etc.), or a combination of hardware and software aspects, collectively referred to herein as a "circuit," "module," or "system."

[0170] The following reference Figure 9 To describe an electronic device 900 according to such an embodiment of the present disclosure. Figure 9 The electronic device 900 shown is merely an example and should not impose any limitation on the functionality and scope of use of the embodiments disclosed herein.

[0171] like Figure 9 As shown, the electronic device 900 is manifested in the form of a general-purpose computing device. The components of the electronic device 900 may include, but are not limited to: at least one processing unit 910, at least one storage unit 920, a bus 930 connecting different system components (including storage unit 920 and processing unit 910), and a display unit 940.

[0172] The storage unit stores program code that can be executed by the processing unit 910, causing the processing unit 910 to perform the steps described in the "Exemplary Methods" section of this specification according to various exemplary embodiments of this disclosure. For example, the processing unit 910 can perform actions such as... Figure 1 Step S110: Construct a first initial training sample set based on the original distribution parameters of the real radome structure model, and construct an adaptive Kriging surrogate model based on the first initial training sample set; Step S120: Calculate the total variance and the first conditional variance of the output failure probability of the real radome structure model based on the adaptive Kriging surrogate model, and calculate the principal sensitivity of the real radome structure model based on the total variance and the first conditional variance; Step S130: Calculate the second conditional variance of the output failure probability based on the adaptive Kriging surrogate model, and calculate the conditional expected value of the output failure probability based on the second conditional variance; Step S140: Calculate the total sensitivity of the real radome structure model based on the second conditional variance and the conditional expected value, and analyze the sensitivity of the real radome structure model based on the principal sensitivity and the total sensitivity.

[0173] Storage unit 920 may include readable media in the form of volatile storage units, such as random access memory (RAM) 9201 and / or cache memory 9202, and may further include read-only memory (ROM) 9203.

[0174] Storage unit 920 may also include a program / utility 9204 having a set (at least one) program module 9205, such program module 9205 including but not limited to: operating system, one or more application programs, other program modules and program data, each or some combination of these examples may include an implementation of a network environment.

[0175] Bus 930 can represent one or more of several types of bus structures, including a memory cell bus or memory cell controller, a peripheral bus, a graphics acceleration port, a processing unit, or a local bus using any of the various bus structures.

[0176] Electronic device 900 can also communicate with one or more external devices 1000 (e.g., keyboard, pointing device, Bluetooth device, etc.), and with one or more devices that enable a user to interact with electronic device 900, and / or with any device that enables electronic device 900 to communicate with one or more other computing devices (e.g., router, modem, etc.). This communication can be performed via input / output (I / O) interface 950. Furthermore, electronic device 900 can also communicate with one or more networks (e.g., local area network (LAN), wide area network (WAN), and / or public networks, such as the Internet) via network adapter 960. As shown, network adapter 960 communicates with other modules of electronic device 900 via bus 930. It should be understood that, although not shown in the figures, other hardware and / or software modules can be used in conjunction with electronic device 900, including but not limited to: microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data backup storage systems.

[0177] From the above description of the embodiments, those skilled in the art will readily understand that the exemplary embodiments described herein can be implemented by software or by combining software with necessary hardware. Therefore, the technical solutions according to the embodiments of this disclosure can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (such as a CD-ROM, USB flash drive, external hard drive, etc.) or on a network, including several instructions to cause a computing device (such as a personal computer, server, terminal device, or network device, etc.) to execute the methods according to the embodiments of this disclosure.

[0178] In exemplary embodiments of this disclosure, a computer-readable storage medium is also provided, on which a program product capable of implementing the methods described above is stored. In some possible implementations, various aspects of this disclosure may also be implemented as a program product including program code that, when the program product is run on a terminal device, causes the terminal device to perform the steps of the various exemplary embodiments of this disclosure described in the "Exemplary Methods" section above.

[0179] The program product for implementing the above-described method according to embodiments of the present disclosure may employ a portable compact disc read-only memory (CD-ROM) and include program code, and may run on a terminal device, such as a personal computer. However, the program product of the present disclosure is not limited thereto. In this document, the readable storage medium may be any tangible medium containing or storing a program that may be used by or in conjunction with an instruction execution system, apparatus, or device.

[0180] The program product may employ any combination of one or more readable media. A readable medium may be a readable signal medium or a readable storage medium. A readable storage medium may be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of readable storage media (a non-exhaustive list) include: an electrical connection having one or more wires, a portable disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof.

[0181] Computer-readable signal media may include data signals propagated in baseband or as part of a carrier wave, carrying readable program code. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. A readable signal medium may also be any readable medium other than a readable storage medium, capable of sending, propagating, or transmitting programs for use by or in conjunction with an instruction execution system, apparatus, or device.

[0182] The program code contained on the readable medium may be transmitted using any suitable medium, including but not limited to wireless, wired, optical fiber, RF, etc., or any suitable combination thereof.

[0183] Program code for performing the operations of this disclosure can be written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Java and C++, and conventional procedural programming languages ​​such as C or similar languages. The program code can execute entirely on the user's computing device, partially on the user's computing device, as a standalone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server. In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0184] Furthermore, the above figures are merely illustrative of the processes included in the method according to exemplary embodiments of this disclosure and are not intended to be limiting. It is readily understood that the processes shown in the above figures do not indicate or limit the temporal order of these processes. Additionally, it is readily understood that these processes may be executed synchronously or asynchronously, for example, in multiple modules.

[0185] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention described herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not invented by this disclosure. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the claims.

Claims

1. A method for analyzing the reliability and sensitivity of distributed parameter uncertainties in a radome structure, characterized in that, include: A first initial training sample set is constructed based on the original distribution parameters of the real radome structure model, and an adaptive Kriging surrogate model is constructed based on the first initial training sample set. This includes: constructing a target function function based on the original model function of the real radome structure model and preset failure conditions; constructing the initial Kriging surrogate model based on the target function function and the first initial training sample set; calculating the U-learning function value of the sampling results included in the first sample pool using the initial Kriging surrogate model; selecting sample points to be updated from the sampling results in the first sample pool based on the U-learning function value; and using the initial Kriging surrogate model as the adaptive Kriging surrogate model when the U-learning function value of the sample point to be updated is greater than or equal to a preset threshold. The total variance and the first conditional variance of the output failure probability of the real radome structure model are calculated based on the adaptive Kriging surrogate model, and the main sensitivity of the real radome structure model is calculated based on the total variance and the first conditional variance. The second conditional variance of the output failure probability is calculated based on the adaptive Kriging surrogate model, and the conditional expected value of the output failure probability is calculated based on the second conditional variance. The total sensitivity of the real radome structure model is calculated based on the second conditional variance value and the conditional expectation value, and the sensitivity of the real radome structure model is analyzed based on the main sensitivity and the total sensitivity.

2. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 1, characterized in that, The first initial training sample set is constructed based on the original distribution parameters of the real radome structure model, including: Obtain the original distribution parameters of the real radome structure model, and use the first target proxy importance sampling probability density function to sample the original input variables to obtain multiple sampling results; A first sample pool is constructed based on the multiple sampling results, and multiple first original sample points are randomly drawn from the first sample pool. Calculate the first distance between the first original sample point and the origin of the real radome structure model, and select multiple first target sample points from the first original sample points based on the first distance; The first buckling response value of each of the first target sample points is calculated using the original model function of the real radome structure model, and the first initial training sample set is constructed based on the first buckling response value.

3. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 2, characterized in that, The original input variables are sampled using the first target surrogate importance sampling probability density function to obtain multiple sampling results, including: Based on the selection principle of the surrogate sampling probability density function, the first original surrogate sampling probability density function of the real radome structure model is determined; In an independent standard normal space, the design points and reliability indices of the real radome structure model are calculated using the improved first-order second-moment method. The first original agent sampling probability density function is shifted from the sampling center to the design point to obtain the first target agent importance sampling probability density function. The original input variables are sampled using the first target proxy importance sampling probability density function to obtain multiple sampling results.

4. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 1, characterized in that, The total variance of the output failure probability of the real radome structure model is calculated based on the adaptive Kriging surrogate model, including: Calculate the first multidimensional distribution parameter variable of the original distribution parameter in an independent standard normal space, and calculate the first variable integration point and the first weight value of the first multidimensional distribution parameter variable; The independent standard normal space is transformed using the integration point of the first variable to obtain the first original model space, and multiple sampling results are selected from the first sample pool to construct the first input variable sample under the first original model space; The first failure domain indicator function is calculated based on the adaptive Kriging proxy model and the first input variable sample. The first failure probability integral point of the output failure probability of the real radome structure model is calculated based on the first target proxy importance sampling probability density function, the first subjective probability density function, the first β-sphere indicator function, the first failure domain indicator function, and the first variable integral point of the first input variable sample. The total variance of the output failure probability is calculated based on the first failure probability integral point and the first weight value.

5. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 1, characterized in that, The first conditional variance of the output failure probability of the real radome structure model is calculated based on the adaptive Kriging surrogate model, including: In the first original model space, calculate the second variable integration point and the second weight value of the second multidimensional distribution parameter variable included in the first original model space, and transform the second variable integration point to the first original model space to obtain the second original model space; Under the second original model space, determine the second original agent sampling probability density function, and determine the second target agent importance sampling probability density function based on the second original agent sampling probability density function; The second input variable sample is extracted from the first sample pool based on the second target agent importance sampling probability density function, and the second failure domain indicator function is calculated based on the second input variable sample and the adaptive Kriging agent model. Calculate the difference between the first multidimensional distribution parameter variable and the second multidimensional distribution parameter variable to obtain the third multidimensional distribution parameter variable, and transform the third variable integration point of the third multidimensional distribution parameter variable to the second original model space to obtain the third original model space; In the third original model space, based on the second input variable sample and the second subjective probability density function, the second β-sphere indicator function, the second failure domain indicator function, and the third variable integration point of the second input variable sample, the second failure probability integration point of the output failure probability of the real radome structure model is calculated. The first conditional expected value of the output failure probability is calculated based on the second failure probability integral point, and the first conditional variance of the output failure probability is calculated based on the first conditional expected value and the third weight value.

6. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 1, characterized in that, The second conditional variance of the output failure probability is calculated based on the adaptive Kriging surrogate model, including: Calculate the fourth variable integration point and the fourth weight value of the third multidimensional distribution parameter variable, and transform the fourth variable integration point to the independent standard normal space to obtain the fourth original model space; In the fourth original model space, the sampling probability density function of the third original agent is determined, and the sampling probability density function of the third target agent is determined based on the sampling probability density function of the third original agent. The third input variable sample is extracted from the first sample pool based on the third target agent important sampling probability density function, and the third failure domain indicator function is calculated based on the second input variable sample and the adaptive Kriging agent model. The second variable integral point of the second multidimensional distribution parameter variable is transformed to the independent standard normal space to obtain the fifth original model space. In the fifth original model space, the third failure probability integral point of the output failure probability is calculated based on the third input variable sample and the third subjective probability density function, the third β ball indicator function, the third failure domain indicator function of the third variable input sample and the second variable integral point. The second conditional variance of the output failure probability is calculated based on the third failure probability integral point and the second weight value of the second multidimensional distribution parameter variable.

7. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 1, characterized in that, The conditional expected value of the output failure probability is calculated based on the second conditional variance value, including: The conditional expected value of the output failure probability is calculated based on the second conditional variance value and the second weight value of the second multidimensional distribution parameter variable.

8. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to claim 1, characterized in that, The method for analyzing the uncertainty, reliability, and sensitivity of the distributed parameters of the radome structure also includes: Obtain the subjective probability density function and the original failure domain indication function of the original distribution parameters of the real radome structure model, and calculate the output failure probability of the real radome structure model based on the subjective probability density function and the original failure domain indication function.

9. The method for analyzing the uncertainty, reliability, and sensitivity of distributed parameters of a radome structure according to any one of claims 1-8, characterized in that, The main sensitivity is used to measure the contribution of any single original distribution parameter to the total variance of the output failure probability. The total sensitivity is used to measure the contribution of any original distributed parameter, acting alone and interacting with other distributed parameters, to the total variance of the output failure probability.

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