A time-varying reliability analysis method based on double-layer BP neural network

Abstract

The application discloses a time-varying reliability analysis method based on a double-layer BP neural network, relates to the field of time-varying structure reliability analysis, and generates a sample pool based on a joint probability density function of random design variables affecting a time-varying structure to be analyzed; calculates sample point weights, screens part of the sample points and establishes a first training data set; constructs an inner-layer BP neural network model and performs training, calculates minimum values of time-varying function functions about time corresponding to each sample point in the first training data set, and forms a second training set; constructs an outer-layer BP neural network model, and performs training by using the second training set; calculates the reliability of the time-varying structure by using the trained outer-layer BP neural network model, solves the minimum values of the time-varying function functions about time through the inner-layer model, and the outer-layer model is constructed based on parameters of the sample points and the minimum values solved by the inner-layer model, and is used for calculating a failure probability, has good applicability, and can be used for time-varying reliability analysis of complex structures.

Description

Technical Field

[0001] This invention relates to the field of time-varying structural reliability analysis technology, and in particular to a time-varying reliability analysis method based on a two-layer BP neural network. Background Technology

[0002] Traditional structural reliability analysis theories, such as first-order second-moment and second-order second-moment methods, usually do not consider time-varying uncertainties such as load and strength degradation. However, in many practical engineering problems, the performance of structures or products often exhibits time-varying characteristics. Therefore, time-varying reliability theories that consider time factors are more in line with the actual situation of engineering.

[0003] Existing time-varying reliability analysis methods using two-layer surrogate models include response surface methodology, Kriging model, and neural network model. The basic idea is to establish a surrogate model of the extrema of the function function with respect to time under given conditions in the inner layer. The extrema of the function function are solved through the inner layer model, transforming the time-varying reliability problem into a time-invariant reliability problem. The outer layer constructs a surrogate model of the input variables and the output extrema. Finally, the failure probability is calculated using the outer layer model. However, this approach fails to effectively focus on the region near the failure surface that has a critical impact on the failure probability, resulting in low sample utilization efficiency and a difficulty in balancing model accuracy and computational efficiency.

[0004] Therefore, a time-varying reliability analysis method based on a two-layer BP neural network is provided to solve the above problems. Summary of the Invention

[0005] To address the aforementioned challenges, this invention provides a time-varying reliability analysis method based on a two-layer BP neural network. The method involves establishing a neural network model of the time variable and function value of the structural system's function function, with the inner layer representing a fixed value for the uncertain variable. This inner model solves for the minimum value of the function function with respect to the time variable. The outer layer constructs a mapping relationship between the uncertain variable and the minimum value of the function function, and calculates the failure probability of the function function. This approach provides relatively accurate calculation results for the reliability assessment of complex structures.

[0006] To achieve the above objectives, this invention provides a time-varying reliability analysis method based on a two-layer BP neural network, comprising the following steps:

[0007] S1: Generate a sample pool based on the joint probability density function of the random design variables that affect the time-varying structure to be analyzed; calculate the weight of each sample point in the sample pool, select some sample points based on the weights, and combine them with the randomly selected sample points to form the first training dataset.

[0008] S2: Construct and train an inner BP neural network model based on the first training dataset. The inner BP neural network model is a proxy model of the function with respect to time when the random design variables take fixed values. S3: Using the trained inner BP neural network model, calculate the minimum value of the time-varying function with respect to time for each sample point in the first training dataset to form the second training set.

[0009] S4: Construct an outer BP neural network model and train it using the second training set; the outer BP neural network model is a model of random design variables and the minimum value of the function.

[0010] S5: The reliability of the time-varying structure is calculated using the trained outer BP neural network model.

[0011] Preferably, S1 specifically includes:

[0012] S11: Generate a sample pool based on the joint probability density function of the random design variables that affect the time-varying structure to be analyzed;

[0013] S12: Calculate and sort the weights of the sample points in the sample pool. Each sample point consists of multiple design variables. Select some sample points based on the weights and combine them with the sample points randomly selected from the sample pool to form the first training dataset.

[0014] Preferably, calculating the weights of sample points in the sample pool specifically includes:

[0015] When the random design variables are independent, the weights of the sample points are expressed as follows:

[0016] ;

[0017] When the random design variables are not independent, the joint probability density function is used to calculate the weights of the sample points, expressed as:

[0018] ;

[0019] in, Let c be the probability density function of the c-th random design variable. Let be the joint probability density function of the (c-1)th and cth variables.

[0020] Preferably, S2 specifically includes:

[0021] S21: For each sample point in the first training dataset, uniform discrete-time observation domain and select several time points; calculate the time-varying function value corresponding to the time point, and combine the random design variable of the sample point with the time point and the corresponding time-varying function value to form the training set of the inner model.

[0022] S22: Using time points and random design variables as inputs and time-varying function values ​​as outputs, establish an inner-layer BP neural network model based on the training set of the inner-layer model constructed in step S21.

[0023] S23: Based on the inner layer model training set, the model accuracy is verified by comparing the mean relative error between the predicted value and the true value of the inner layer BP neural network model. If the accuracy requirement is not met, the sample points are added to the inner layer model training set, and steps S22 and S23 are repeated for iterative training until the accuracy requirement is met, and the trained inner layer BP neural network model is obtained.

[0024] Preferably, the mean relative error is expressed as:

[0025] ;

[0026] in, and Corresponding to the first The model's predicted value and the actual value of the function for each sample point. m The number of samples used in the error assessment. To design variables for randomization, The time point is the h-th sample point.

[0027] Preferably, S3 specifically includes:

[0028] S31: Using the trained inner-layer BP neural network model, calculate the minimum value of the time-varying function of each sample point in the first training dataset with respect to time;

[0029] S32: The second training set is formed based on the sample points in the first training dataset and the minimum values ​​calculated by the corresponding inner BP neural network model.

[0030] Preferably, S4 specifically includes:

[0031] S41: Using random design variables from the second training set as input and time-varying function values ​​as output, construct an outer BP neural network model;

[0032] S42: Input the sample points in the sample pool into the outer BP neural network model to obtain several output values; select several values ​​whose absolute values ​​are less than a preset value from the output values, and calculate the minimum value of the function function with respect to time at the corresponding sample point; evaluate the mean relative error between the minimum value of the function function and the predicted value of the outer model; if the mean error does not meet the preset accuracy requirement, add the above sample points and their true minimum value of the function function to the second training dataset, and repeat steps S41 and S42 for iterative training until the model accuracy meets the requirements, and obtain the trained outer BP neural network model.

[0033] Preferably, S5 specifically includes:

[0034] S51: Input the data from the sample pool into the trained outer BP neural network model to calculate the failure probability of the time-varying structure to be analyzed.

[0035] S52: Based on the failure probability of the time-varying structure to be analyzed, calculate the coefficient of variation of the failure probability; determine whether the coefficient of variation is less than a preset threshold; if it is less than the threshold, the calculation result is reliable; if it is not less than the threshold, expand the capacity of the sample pool and return to step S51 to recalculate the failure probability.

[0036] Preferably, the failure probability in S51 is expressed as:

[0037] ;

[0038] in, For the time-varying failure domain indicator function, the outer BP neural network model hour, Otherwise, it is 0. For capacity The sample pool.

[0039] Preferably, the failure probability in S52 is expressed as:

[0040] .

[0041] Therefore, this invention employs a time-varying reliability analysis method based on a two-layer BP neural network. The BP neural network model has good applicability in calculating time-varying reliability problems. Through the synergistic effect of multiple hidden layers and neurons, it can effectively capture complex nonlinear mapping relationships and has the ability to approximate arbitrarily complex nonlinear functions, providing relatively accurate calculation results for the reliability assessment of complex structures. Furthermore, the inner layer constructs a BP neural network model of the structural system's function function with respect to time variables under the condition that the uncertainty variable takes a fixed value, using this model to obtain the minimum value of the function function with respect to time. The outer layer utilizes the efficient and high-precision fitting capability of the BP neural network to construct a mapping relationship between the uncertainty variable and the minimum value of the function function. By using sample point selection techniques, the fitting efficiency of the neural network is improved, demonstrating good applicability and suitability for time-varying reliability analysis of complex structures.

[0042] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating a time-varying reliability analysis method based on a two-layer BP neural network in this invention.

[0044] Figure 2 This is a schematic diagram of a four-bar linkage in an embodiment of the present invention. Detailed Implementation

[0045] The following detailed description of embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0046] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0047] The terms "comprising" or "including" as used in this invention mean that the element preceding the term encompasses the element listed after the term, and do not exclude the possibility of encompassing other elements. Terms such as "inner," "outer," "upper," and "lower" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. When the absolute position of the described object changes, the relative positional relationship may also change accordingly. In this invention, unless otherwise explicitly specified and limited, the term "attached" and similar terms should be interpreted broadly. For example, it can refer to a fixed connection, a detachable connection, or an integral part; it can refer to a direct connection or an indirect connection through an intermediate medium; it can refer to the internal communication of two elements or the interaction relationship between two elements. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0048] Example

[0049] A time-varying reliability analysis method based on a two-layer BP neural network, such as Figure 1 As shown, it includes the following steps:

[0050] S1: Generate a sample pool based on the joint probability density function of the random design variables that affect the time-varying structure to be analyzed; calculate the weight of each sample point in the sample pool, select some sample points based on the weights, and combine them with the randomly selected sample points to form the first training dataset.

[0051] S11: Generate a sample pool based on the joint probability density function of the random design variables that affect the time-varying structure to be analyzed; generate the sample pool.

[0052] Specifically, based on the input variables The joint probability density function has a generation capacity of sample pool ;

[0053] S12: Calculate and sort the weights of the sample points in the sample pool. Each sample point consists of multiple design variables. Select some sample points based on the weights and combine them with the sample points randomly selected from the sample pool to form the first training dataset.

[0054] Specifically, the weights of the sample points are calculated and sorted, for each sample point. It consists of multiple design variables, denoted as , Given the number of design variables, and when the design variables are independent, the weight of the sample points is:

[0055] ;

[0056] When the design variables are not independent, the calculation is performed based on the joint probability density function, i.e.:

[0057] ;

[0058] In the formula, Let be the probability density function of the c-th random variable. Let be the joint probability density function of the (c-1)th and cth variables.

[0059] Select the sample points with the lowest weights, and randomly select a certain number of sample points from the sample pool to form a group of samples with the following size. dataset .

[0060] S2: Construct and train an inner BP neural network model based on the first training dataset. The inner BP neural network model is a proxy model of the function with respect to time when the random design variable takes a fixed value.

[0061] S21: For each sample point in the first training dataset, uniform discrete-time observation domain and select several time points; calculate the time-varying function value corresponding to the time point, and combine the random design variable of the sample point with the time point and the corresponding time-varying function value to form the training set of the inner model.

[0062] Specifically, in the dataset Select an input variable Uniform discrete-time observation domain for Each time point is denoted as... ,exist Random selection Each time point is denoted as... Calculate the corresponding time-varying function value and compare it with the selected input variable. Merge into training set , .

[0063] S22: Using time points and random design variables as inputs and time-varying function values ​​as outputs, establish an inner-layer BP neural network model based on the training set of the inner-layer model constructed in step S21.

[0064] Specifically, construct a BP neural network model. Based on the training set... The data in the data is presented in terms of time points. and design variables As input, the function value As the output, an inner BP neural network model is established, denoted as... ;

[0065] S23: Based on the inner layer model training set, the model accuracy is verified by comparing the mean relative error between the predicted value and the true value of the inner layer BP neural network model. If the accuracy requirement is not met, the sample points are added to the inner layer model training set, and steps S22 and S23 are repeated for iterative training until the accuracy requirement is met, and the trained inner layer BP neural network model is obtained.

[0066] Specifically, the accuracy of the inner BP neural network model is verified. This process involves using the inner BP neural network model to calculate the input variables. Time sample pool Corresponding predicted value ,choose The absolute value is less than of A number, denoted as The function is called to calculate the corresponding time point (i.e., the point near the failure surface). The actual value of the function ;

[0067] The error between the true value and the output value of the inner BP neural network model is calculated for accuracy verification, and the mean of the relative error is selected. As an indicator:

[0068] ;

[0069] in, and Corresponding to the first The output value of the inner layer BP neural network model and the true value of the function function for each sample point, where m is the number of samples participating in the error evaluation. The smaller the value, the higher the accuracy of the model. This embodiment allows for a mean relative error. .

[0070] If the error meets the accuracy requirements, the training of the inner BP neural network can be considered complete; otherwise, it will be considered complete. and their corresponding function values Add to training set In this process, the inner BP neural network model is iterated and the training process is repeated until the model's accuracy meets the requirements, thus obtaining the trained inner BP neural network model. .

[0071] S3: Using the trained inner-layer BP neural network model, calculate the minimum value of the time-varying function with respect to time for each sample point in the first training dataset to form the second training set;

[0072] S31: Using the trained inner-layer BP neural network model, calculate the minimum value of the time-varying function of each sample point in the first training dataset with respect to time;

[0073] Specifically, based on the inner BP neural network model obtained from S2 The input parameters are calculated as follows: The time-varying function has a minimum value with respect to time. S2 should be repeated. This process is repeated to obtain the minimum value of the function function with respect to time for each selected sample point, denoted as... ;

[0074] S32: The second training set is formed based on the sample points in the first training dataset and the minimum values ​​calculated by the corresponding inner BP neural network model.

[0075] Specifically, using the selected sample points and the minimum value calculated by the corresponding inner model. The training set that makes up the outer neural network That is, the second training set.

[0076] S4: Construct an outer BP neural network model and train it using the second training set; the outer BP neural network model is a model of random design variables and the minimum value of the function.

[0077] S41: Using random design variables from the second training set as input and time-varying function values ​​as output, construct an outer BP neural network model;

[0078] Specifically, using the training set In the middle, sample points The parameters are used as inputs, and the time-varying function minimization value corresponding to the sample points is obtained. As output, construct the outer BP neural network model. ;

[0079] S42: Input the sample points in the sample pool into the outer BP neural network model to obtain several output values; select several values ​​whose absolute values ​​are less than a preset value from the output values, and calculate the minimum value of the function function with respect to time at the corresponding sample point; evaluate the mean relative error between the minimum value of the function function and the predicted value of the outer model; if the mean error does not meet the preset accuracy requirement, add the above sample points and their true minimum value of the function function to the second training dataset, and repeat steps S41 and S42 for iterative training until the model accuracy meets the requirements, and obtain the trained outer BP neural network model.

[0080] Specifically, the sample pool Each sample point is fed into the outer BP neural network model. The calculation yields a total of Each output value is denoted as... ;exist Select the absolute value less than of Number of values Execute S2 to calculate The corresponding sample points (i.e., points near the failure surface). The minimum value of the function with respect to time. ; Evaluate the minimum value of the function Compared with the predicted values ​​of the outer BP neural network model To determine the error, choose the mean of the relative error ( As an indicator, the calculation formula is shown in S23; if the error meets the requirements, then the outer BP neural network model... The training is complete; otherwise, the verification point will be checked. The parameters and corresponding minimum values ​​of the functional functions Add to training set In this process, the outer BP neural network model is iteratively upgraded, and the training process is repeated until the accuracy meets the requirements, resulting in the trained outer BP neural network model. ;

[0081] S5: The reliability of the time-varying structure is calculated using the trained outer BP neural network model.

[0082] S51: Input the data from the sample pool into the trained outer BP neural network model to calculate the failure probability of the time-varying structure to be analyzed.

[0083] Specifically, the capacity is sample pool Input into outer model In the calculation of the failure probability of the structure :

[0084] ;

[0085] In the formula, This is a time-varying failure domain indicator function. hour, Otherwise, it is 0.

[0086] S52: Based on the failure probability of the time-varying structure to be analyzed, calculate the coefficient of variation of the failure probability; determine whether the coefficient of variation is less than a preset threshold; if it is less than the threshold, the calculation result is reliable; if it is not less than the threshold, expand the capacity of the sample pool and return to step S51 to recalculate the failure probability.

[0087] Calculate the probability of failure The coefficient of variation reflects the deviation between the output failure probability and the true value. The smaller the deviation, the higher the reliability of the result.

[0088] ;

[0089] This embodiment adopts As a test criterion, if If so, it means the calculation result is reliable. This proves that the sample size used to calculate the failure probability is too small, and it is necessary to increase the sample size and recalculate the failure probability.

[0090] Example 1

[0091] To further verify the effectiveness of the method presented in this paper, we selected a four-bar linkage mechanism, which is commonly used in time-varying reliability analysis, as an example. We focused on whether the error of the mechanism's output function exceeded a given threshold. If it did not exceed the threshold, the mechanism was reliable; otherwise, it failed.

[0092] like Figure 2 The four-bar linkage shown has the following geometric dimensional variables for its four bars: , The means of the distributions are respectively , , , The standard deviation is All ...

[0093] ;

[0094] in, Considered as input to the motion process, and It is considered as the output of the motion process.

[0095] and The expression can be solved based on the above formula as follows:

[0096] ;

[0097] in, , , .

[0098] This embodiment primarily considers output. The time-varying output function can be obtained as follows:

[0099] ;

[0100] The key concern of this four-bar linkage is whether the error of the output function exceeds a given threshold. If the threshold is not exceeded, the mechanism is reliable; otherwise, the mechanism is faulty. Treat as a time variable The performance response function is established as follows:

[0101] ;

[0102] set up The observation domain is Set error threshold The value is 0.8, which is a fixed value. The calculation results are shown in Table 1.

[0103] Table 1 Comparison of Results from Different Algorithms

[0104] ;

[0105] The Monte Carlo method was used to generate 218 sample points, and the time domain was uniformly discretized into 101 time points. A total calculation was required. The secondary function, calculated as 2.30 × 10⁻⁶, yields a result of 2.30 × 10⁻⁶. -3 Using this method, with 2317 function calls, the failure probability is 2.32 × 10⁻⁶. -3 The error between the two methods is 0.8%. As can be seen from the table, this method can improve computational efficiency while ensuring computational accuracy.

[0106] Therefore, the present invention adopts the above-mentioned time-varying reliability analysis method based on a two-layer BP neural network. The BP neural network model has good applicability in calculating time-varying reliability problems. Through the synergistic effect of multiple hidden layers and neurons, it can effectively capture complex nonlinear mapping relationships and has the ability to approximate any complex nonlinear function. It can provide relatively accurate calculation results for the reliability assessment of complex structures and has good applicability. It can be used for time-varying reliability analysis of complex structures.

[0107] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A time-varying reliability analysis method based on a two-layer BP neural network, characterized in that, Includes the following steps: S1: Generate a sample pool based on the joint probability density function of the random design variables that affect the time-varying structure to be analyzed; Calculate the weight of each sample point in the sample pool, select some sample points based on the weights, and combine them with randomly selected sample points to form the first training dataset. S2: Construct and train an inner BP neural network model based on the first training dataset. The inner BP neural network model is a proxy model of the function with respect to time when the random design variables take fixed values. S3: Using the trained inner BP neural network model, calculate the minimum value of the time-varying function with respect to time for each sample point in the first training dataset to form the second training set. S4: Construct an outer BP neural network model and train it using the second training set; the outer BP neural network model is a model of random design variables and the minimum value of the function. S5: The reliability of the time-varying structure is calculated using the trained outer BP neural network model; When the time-varying structure is a four-bar linkage, the geometric dimensions of its four bars vary as follows: , The means of the distributions are respectively , , , The standard deviation is All ... ； in, As input for the motion process, and The output of the motion process; and The expression is solved according to the above formula as follows: ； in, , , ; Consider the output The time-varying output function is obtained as follows: ； Will Treat as a time variable The performance response function is established as follows: 。 2. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 1, characterized in that: S1 specifically includes: S11: Generate a sample pool based on the joint probability density function of the random design variables that affect the time-varying structure to be analyzed; S12: Calculate and sort the weights of the sample points in the sample pool. Each sample point consists of multiple design variables. Select some sample points based on the weights and combine them with the sample points randomly selected from the sample pool to form the first training dataset.

3. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 2, characterized in that: The calculation of the weights of sample points in the sample pool specifically includes: When the random design variables are independent, the weights of the sample points are expressed as follows: ； When the random design variables are not independent, the joint probability density function is used to calculate the weights of the sample points, expressed as: ； in, Let c be the probability density function of the c-th random design variable. Let be the joint probability density function of the (c-1)th and cth variables.

4. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 3, characterized in that: S2 specifically includes: S21: For each sample point in the first training dataset, uniform discrete-time observation domain and select several time points; calculate the time-varying function value corresponding to the time point, and combine the random design variable of the sample point with the time point and the corresponding time-varying function value to form the training set of the inner model. S22: Using time points and random design variables as inputs and time-varying function values ​​as outputs, establish an inner-layer BP neural network model based on the training set of the inner-layer model constructed in step S21. S23: Based on the inner layer model training set, the model accuracy is verified by comparing the mean relative error between the predicted value and the true value of the inner layer BP neural network model. If the accuracy requirement is not met, the sample points are added to the inner layer model training set, and steps S22 and S23 are repeated for iterative training until the accuracy requirement is met, and the trained inner layer BP neural network model is obtained.

5. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 4, characterized in that: The mean relative error is expressed as: ； in, and Corresponding to the first The model's predicted value and the actual value of the function for each sample point. m The number of samples used in the error assessment. To design variables for randomization, The time point is the h-th sample point.

6. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 1, characterized in that: S3 specifically includes: S31: Using the trained inner-layer BP neural network model, calculate the minimum value of the time-varying function of each sample point in the first training dataset with respect to time; S32: The second training set is formed based on the sample points in the first training dataset and the minimum values ​​calculated by the corresponding inner BP neural network model.

7. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 6, characterized in that: S4 specifically includes: S41: Using random design variables from the second training set as input and time-varying function values ​​as output, construct an outer BP neural network model; S42: Input the sample points in the sample pool into the outer BP neural network model to obtain several output values; select several values ​​whose absolute values ​​are less than a preset value from the output values, and calculate the minimum value of the function function with respect to time at the corresponding sample point; evaluate the mean relative error between the minimum value of the function function and the predicted value of the outer model; if the mean error does not meet the preset accuracy requirement, add the above sample points and their true minimum value of the function function to the second training dataset, and repeat steps S41 and S42 for iterative training until the model accuracy meets the requirements, and obtain the trained outer BP neural network model.

8. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 1, characterized in that: S5 specifically includes: S51: Input the data from the sample pool into the trained outer BP neural network model to calculate the failure probability of the time-varying structure to be analyzed. S52: Based on the failure probability of the time-varying structure to be analyzed, calculate the coefficient of variation of the failure probability; determine whether the coefficient of variation is less than a preset threshold; if it is less than the threshold, the calculation result is reliable; if it is not less than the threshold, expand the capacity of the sample pool and return to step S51 to recalculate the failure probability.

9. The time-varying reliability analysis method based on a two-layer BP neural network as described in claim 8, characterized in that: The failure probability in S51 is expressed as follows: ； in, For the time-varying failure domain indicator function, the outer BP neural network model hour, Otherwise, it is 0. For capacity The sample pool.

10. The time-varying reliability analysis method based on a two-layer BP neural network according to claim 9, characterized in that: The failure probability in S52 is expressed as follows: 。

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