Active learning reliability analysis method for mirror support system

Abstract

The present invention belongs to the field of reliability analysis technology and relates to an active learning reliability analysis method for a reflector support system, which realizes the solution of the failure probability of the reflector system. First, the sampling boundary is determined based on the mean and standard deviation of the flexible node radius and the flexible node distribution circle diameter in the reflector support system, and the Latin hypercube sampling method is used to generate initial training data and uniform samples; a Kriging proxy model is constructed to obtain the predicted mean and predicted variance of the optical performance function; the U function is used as the active learning criterion; it is determined whether the candidate sample set is empty, and if so, the method is terminated, and then the failure probability of the reflector support system is calculated using the Kriging model. Otherwise, the sample with the largest predicted variance is selected from the candidate sample set, and a one-dimensional variable is introduced to represent the change of the sample along the gradient direction of the optical performance function. A new objective function is constructed, and the calculated new sample is added to the training data set, and the process is repeated until the candidate sample set is empty.

Description

Technical Field

[0001] This invention belongs to the field of reliability analysis technology, and relates to an active learning reliability analysis method for a mirror support system. Background Art

[0002] In optical systems, the reliability of the mirror support system plays a crucial role in the performance of optical equipment. As a core component, the stability of the support structure of a large-aperture mirror directly affects key performance indicators of the optical system, such as imaging quality and beam collimation. Due to uncertainties in manufacturing processes, material properties, and other factors, mirror support systems face reliability challenges in actual operation.

[0003] Traditional reliability analysis methods, such as those described in patent CN119049611B, have several shortcomings when dealing with complex structures like mirror support systems. Decomposition-based reliability analysis methods approximate performance functions, leading to significant errors when handling highly nonlinear performance functions. While Monte Carlo sampling methods can theoretically yield accurate results, they require a large number of samples to calculate small failure probabilities, resulting in extremely high computational costs. Surrogate model-based reliability analysis methods also have limitations when handling small failure probabilities in mirror support systems. The samples used for active learning are the same as those used for failure probability calculation, leading to uneven distribution of samples around the limiting state function, making it difficult to accurately identify failure regions, and consequently affecting the efficiency and accuracy of reliability analysis. Summary of the Invention

[0004] To address the problem that existing reliability analysis methods cannot handle low failure probabilities in mirror support systems, this invention provides an active learning reliability analysis method for mirror support systems, enabling the calculation of failure probabilities for mirror systems, and aiming to improve the accuracy and efficiency of low failure probability reliability analysis for mirror support systems.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] An active learning reliability analysis method for a mirror support system includes the following steps:

[0007] Step 1: Determine the random parameters and their distributions of the inner and outer ring distribution radii, inner and outer flexible joint radii, and the mirror height of the mirror support system. Based on the mean μ and standard deviation σ of the random variables of the inner and outer ring distribution radii and the mirror height in the mirror support system, determine the sampling boundary vector as follows:

[0008] x l ,x u =μ±4σ(1)

[0009] Where, x l and x u The upper and lower bounds of the sampling boundary are defined. Within this boundary, an initial training dataset T is generated using the Latin hypercube sampling method, containing the inner circle distribution radius, outer circle distribution radius, inner circle flexible joint radius, outer circle flexible joint radius, and mirror height, to meet the space filling requirements. Simultaneously, within this boundary, independent datasets N are generated using the Latin hypercube sampling method. uni A dataset S, consisting of uniformly distributed samples of the inner and outer circle distribution radii, the inner and outer flexible joint radii, and the mirror height, is used for active learning. Unlike traditional methods, the uniformly distributed sample dataset S is uniformly distributed within the boundary, which can more effectively cover the limit state function region, and the number of uniform samples N is [not specified]. uni This can be adjusted according to the actual situation. By calling the simulation model of the reflector support system, the response value of the initial training dataset T is calculated and used as the output of the training data.

[0010] Step 2: Define the optical performance function required for the reliability analysis of the mirror support system as G(x), and construct a Kriging surrogate model using the training dataset. Obtain the mean predicted optical performance function of the Kriging model at any point x within the sampling boundary. and prediction variance for:

[0011]

[0012] and

[0013]

[0014] in, and Here are the parameters of the Kriging model, and 1 represents a vector with all elements equal to 1. R is the correlation matrix, f is the training data for the optical performance function, and r(x) is the correlation vector between point x and the training points.

[0015] Step 3: Definition The limit state function, U-function, is used as the active learning criterion and is expressed as:

[0016]

[0017] Calculate the U(x) value for each sample in the uniform sample set. Select samples that satisfy U(x) ≤ 2 to form a candidate sample set S. candi This method ensures that the selected samples are both close to the limiting state function and that the training data is more evenly distributed.

[0018] Step 4: Determine the candidate sample set S candiIf the value is empty, the algorithm terminates and then uses the Kriging model to calculate the failure probability of the mirror support system using Monte Carlo simulation. Otherwise, proceed to step five.

[0019] Step 5: From the candidate sample set S candi Select the sample x with the largest prediction variance candi Since this sample is typically not on the LSF, a local refinement method is used for optimization. A one-dimensional variable λ is introduced to represent sample x. candi Construct a new objective function by varying along the gradient direction of the optical performance function:

[0020]

[0021] in, For the optical performance function in x candi The gradient at a given point is based on the standard deviation σ of each dimension of the random variable. i Determine the optimal radius r:

[0022]

[0023] Optical performance function The values ​​in the i-th dimension, i = 1 to 5, represent five random variables. The local refinement problem is transformed into an optimization problem of minimizing the objective function within the range λ ∈ [-r, r]. This is solved using a sequential quadratic programming algorithm.

[0024]

[0025] The optimized λ is obtained * Then, new samples are calculated:

[0026]

[0027] Point x new Add to the training dataset, repeat steps two through five, until S in step four. candi If the value is empty, the algorithm terminates.

[0028] The effective gain effect of the present invention is as follows:

[0029] By separating the samples used for failure probability calculation and active learning, the use of uniform samples makes the training data more evenly distributed around the LSF, reducing the number of samples required to identify failure regions and lowering computational costs. When dealing with complex structures such as mirror support systems, the computation time is significantly reduced compared to traditional methods. The local refinement process makes candidate samples closer to the LSF, improving the Kriging model's approximation accuracy of the LSF and thus enhancing the accuracy of failure probability calculation. In various numerical examples and real-world engineering cases, the failure probability results calculated by the method of this invention are closer to the true values. Attached Figure Description

[0030] Figure 1 This is a flowchart illustrating the reliability analysis of the mirror support system of the present invention.

[0031] Figure 2 This is a schematic diagram of a reflector support system according to an embodiment of the present invention; in the figure, 1 is the mirror chamber, 2 is the reflector, and 3 is the reflector support column.

[0032] Figure 3 This section describes the variables of the mirror support system in an embodiment of the present invention.

[0033] Figure 4 This is the mesh division result of the mirror support system in an embodiment of the present invention.

[0034] Figure 5 The results of finite element deformation analysis of the mirror support system in this embodiment of the invention are shown. Detailed Implementation

[0035] The present invention will be further described below with reference to specific embodiments.

[0036] like Figure 1 As shown, this invention provides an active learning reliability analysis method for a main mirror support system. First, the variable range and performance function of the reliability analysis problem of the main mirror support system are obtained. Then, Latin hypercube sampling is used to obtain an initial training set and an initial uniform sample set, and the training set is evaluated. Next, a Kriging surrogate model is established for the performance function, and a U function is used to select a candidate sample set. The loop is exited if the candidate sample set is empty, and the sample with the largest prediction variance is selected as the candidate sample. Local refinement is used for optimization to obtain new training points, which are then evaluated and added to the training set. A new Kriging model is then constructed, and the above steps are repeated until the algorithm meets the stopping criterion. The results of the reliability analysis of the main mirror support system are output.

[0037] Figure 2The structure of the reflector support system consists of a mirror chamber 1, a reflector 2, and a reflector support column 3. The mirror chamber 1 is the fixing device for the reflector system, used to maintain the stability of the reflector 2. One end of the reflector support column 3 is fixed to the mirror chamber 1, and the other end is fixed to the reflector 2. The reflector 2 is made of glass, and the reflector support column 3 is made of steel. Figure 3 The parameters of the reflector support system are described in Table 1, and the random distribution of each parameter is shown in Table 1.

[0038] Table 1 Random variables of the mirror support system

[0039]

[0040] The performance of an optical system depends to some extent on the imaging quality of the mirror, especially its resolving power. This embodiment uses the Rayleigh criterion, which defines the minimum angular distance at which two point light sources can be distinguished as independent objects. The Rayleigh criterion states that if the wavefront deviation is less than one-quarter of the light wavelength, the performance of the optical system will not be significantly affected. Therefore, the performance function is defined as:

[0041]

[0042] Where λ is the wavelength of light, set to 1064 nm. W Wavefront deviation can be calculated based on the deformation of the reflector. However, the reflector deformation cannot be calculated directly; finite element analysis is required to obtain the deformation data. Therefore, the commercial software ANSYS Workbench was used. The mesh generation for the finite element analysis of the reflector support system is as follows: Figure 4 As shown, the deformation distribution obtained through finite element calculation is as follows: Figure 5 As shown.

[0043] First, the 5-dimensional design variables are sampled using the Latin hypercube sampling method to obtain the initial training dataset T, with an initial training size of 20. Then, the same Latin hypercube sampling method is used to generate a dataset S consisting of 2000 uniform samples for active learning. The response values ​​of the initial training dataset T are calculated using the mirror support system simulation model, and this is used as the output of the training data. Finally, a Kriging surrogate model is constructed using the training data to evaluate the performance function.

[0044] Then, using the U function as the active learning criterion, samples satisfying U(x)≤2 are selected to form a candidate sample set S. candi Determine the sample set S. candi If the value is empty, the algorithm terminates and then uses the Kriging model to calculate the failure probability of the mirror support system using Monte Carlo simulation. Otherwise, it starts from the candidate sample set S. candi Select the sample x with the largest prediction variance candiAnd a local refinement method is used for optimization to obtain a new sample x. new .

[0045] Finally, point x new Add to the training dataset to build the Kriging model, and repeat the above process until the sample set S is reached. candi Empty. The reliability analysis result is 6.800 × 10⁻⁶. -5 It took 9175.173 seconds. Compared to AK-MCS's 59276.74 seconds, the computational efficiency has been greatly improved.

Claims

1. An active learning reliability analysis method for a mirror support system, characterized in that, The steps include: Step 1: Determine the random parameters and their distributions of the inner and outer ring distribution radii, inner and outer flexible joint radii, and the mirror height of the mirror support system. Based on the mean μ and standard deviation σ of the random variables of the inner and outer ring distribution radii and the mirror height in the mirror support system, determine the sampling boundary vector as x. l ,x u =μ±4σ(1) Among them, x l and x u The upper and lower bounds of the sampling boundary are defined. Within this boundary, an initial training dataset T is generated using the Latin hypercube sampling method, containing the inner circle distribution radius, outer circle distribution radius, inner circle flexible joint radius, outer circle flexible joint radius, and mirror height, to meet the space filling requirements. Simultaneously, within this boundary, independent datasets N are generated using the Latin hypercube sampling method. uni A dataset S, consisting of uniform samples of inner circle distribution radius, outer circle distribution radius, inner circle flexible joint radius, outer circle flexible joint radius, and mirror height, is used for active learning. The response value of the initial training dataset T is calculated by calling the mirror support system simulation model, and is used as the output of the training data. Step 2: Define the optical performance function required for the reliability analysis of the mirror support system as G(x), and construct a Kriging surrogate model using the training dataset; obtain the mean predicted optical performance function of the Kriging model at any point x within the sampling boundary. and prediction variance for: and in, and Here are the parameters of the Kriging model, 1 is a vector with all elements equal to 1; R is the correlation matrix, f is the training data for the optical performance function, and r(x) is the correlation vector between point x and the training point. Step 3: Definition The limit state function, U-function, is used as the active learning criterion and is expressed as: Calculate the U(x) value for each sample in the uniform sample set; select samples that satisfy U(x) ≤ 2 to form a candidate sample set S. candi ; Step 4: Determine the candidate sample set S candi If the value is empty, the algorithm terminates and then uses the Kriging model to calculate the failure probability of the mirror support system using the Monte Carlo simulation method; otherwise, proceed to step five. Step 5: From the candidate sample set S candi Select the sample x with the largest prediction variance candi Since this sample is usually not on the LSF, a local refinement method is used for optimization; a one-dimensional variable λ is introduced to represent sample x. candi Construct a new objective function by varying along the gradient direction of the optical performance function: in, For the optical performance function in x candi The gradient at a given point is based on the standard deviation σ of each dimension of the random variable. i Determine the optimal radius r: Optical performance function The values ​​in the i-th dimension, i = 1 to 5, represent five random variables. The local refinement problem is transformed into an optimization problem of minimizing the objective function within the range λ ∈ [-r, r]. This problem is solved using a sequential quadratic programming algorithm. The optimized λ is obtained * Then, a new sample is calculated: Point x new Add to the training dataset, repeat steps two through five, until S in step four. candi If the value is empty, the algorithm terminates.

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