A time-varying reliability analysis method based on neural network extreme value response surface

Abstract

The application provides a time-varying reliability analysis method based on a neural network extreme response surface, and relates to an aero-engine transmission system, and comprises the following steps: 1, time-varying random process discretization of the transmission system; 2, sampling based on the inverse reliability analysis principle; 3, solving the extreme value of the limit state function of the transmission system for the sampling data; 4, constructing the extreme response surface of the limit state function of the transmission system based on the neural network response surface; 5, constructing the inverse reliability analysis model by using the extreme response surface of the transmission system; and 6, solving the constructed inverse reliability analysis model of the transmission system by using a genetic algorithm. The critical failure points of the state function at each time point are found out by discretizing the time, so that the time-varying reliability analysis of the transmission system is converted into a conventional reliability problem for solving. The calculation efficiency of the time-varying reliability of the aero-engine transmission system is greatly improved, so that the design efficiency and design quality of the aero-engine are effectively improved, and the design cost is reduced.

Description

TECHNICAL FIELD

[0001] The application provides an aero-engine transmission system time-varying reliability analysis method based on a neural network extreme value response surface, and belongs to the field of complex product multidisciplinary reliability analysis. BACKGROUND

[0002] Various uncertainties exist in engineering design, such as manufacturing errors, material property differences, etc., which are reflected as uncertainties of design variables. If these uncertainties are not considered in design, the product reliability of the design is likely to be very low, and the performance is unstable. For precision engineering systems and products, a small error can cause irreparable loss, and the consequences are unpredictable. Traditional reliability analysis, also known as static reliability analysis or time-invariant reliability analysis, is characterized by a fixed failure probability. However, due to the influence of time-varying uncertainties, such as corrosion degradation and dynamic load loading, the failure degree of the product often changes with time, so the reliability analysis considering time-varying uncertainties is also called time-varying reliability analysis.

[0003] Compared with traditional reliability analysis, complex product design considering time-varying uncertainties adds a time dimension to the reliability constraint condition. Due to the existence of time-varying factors, the traditional reliability analysis method is difficult to directly solve the reliability of the state function. The reliability analysis problem considering time-varying uncertainties mainly includes two categories: one is that the state function has time-varying characteristics, and the other is that the design variables or parameters have time-varying characteristics. Among them, the first type of time-varying reliability problem is more general and has practical application significance, because the time-varying uncertainties in engineering design are mainly caused by the time dimension of failure mode changes such as part size wear, material aging, corrosion and random load. It is more reasonable to use a state function containing time factors to represent the reliability problem. Therefore, the application proposes a time-varying reliability analysis method based on a neural network extreme value response surface for the first type of time-varying reliability analysis problem.

[0004] Due to the addition of the time dimension, the difficulty of time-varying reliability analysis is doubled. The commonly used methods for solving time-varying reliability are Monte Carlo method and crossing rate method, the former has slow solving speed, and the latter has low calculation accuracy, so the above methods have limited application in time-varying reliability research. Considering that time-invariant reliability analysis methods have been relatively mature today, it will be more efficient if time-varying reliability can be converted into time-invariant reliability for solving. Therefore, the application draws lessons from the principle of limit value response surface, and converts the time-varying reliability problem into a conventional reliability problem for solving.

[0005] The application adopts a response surface method to approximately consider the extreme value of a limit state function of a time-varying factor, finds out critical failure points of the state function at various time points through discretization of time according to an extreme value response principle, and thus converts time-varying reliability analysis into a conventional reliability problem for solving. The reliability analysis research considering time-varying characteristics is of great significance to improve the reliability design level and the reliability management level of an aero-engine transmission system. SUMMARY

[0006] (I) Objectives of the application

[0007] The application aims to provide an aero-engine time-varying reliability solving method based on a neural network extreme value response surface, which converts a time-varying reliability problem into a conventional reliability problem for solving by using an extreme value response surface principle. The overall flow of the method is shown in FIG. Figure 1 .

[0008] (II) Technical solutions

[0009] The application introduces an aero-engine transmission system time-varying reliability analysis method based on a neural network extreme value response surface, which includes the following steps:

[0010] Step one, discretization of a transmission system time-varying random process;

[0011] Step two, sampling based on an inverse reliability analysis principle;

[0012] Step three, solving an extreme value of a transmission system limit state function for sampling data;

[0013] Step four, constructing an extreme value response surface of the transmission system limit state function based on a neural network response surface;

[0014] Step five, constructing an inverse reliability analysis model by using the transmission system extreme value response surface;

[0015] Step six, solving the constructed transmission system inverse reliability analysis model by using a particle swarm algorithm;

[0016] To make the technical solutions of the application more detailed and clear, the following is described in detail, please refer to FIG. Figure 1 .

[0017] In step one, the discretization of the transmission system time-varying random process is specifically as follows:

[0018] For the general aero-engine transmission system limit state function g(x, y(t), t) containing time-varying factors, since the random process y(t) cannot be directly processed, discretization is needed to simplify the calculation amount. For the convenience of calculation, it is assumed that all random processes are Gaussian random processes, and all random variables follow normal distribution. The optimal linear estimation expansion method is used to discretize the Gaussian random process. The specific implementation process is as follows:

[0019] Let the mean and variance of the Gaussian random process y(t) be μ Y (t) and σ(t), and the optimal linear estimation expansion method is used to discretize the Gaussian random process. Assuming that the random process is discretized into P time periods within the time period T, y(t) can be expressed as:

[0020]

[0021] In the formula, ζ i is an independent normal random variable; λ i and Φ i are the eigenvalues and eigenvectors of the correlation coefficient matrix C, where the eigenvalues are arranged in descending order, and the expression of the correlation matrix is C i,j ={ρ X (t i ,t j ), i, j = 1, 2,..., P}, ρ X (t i ,t j ) is the probability density from t i to t j , C i,j is the set of probability densities of P time periods within the time period T; and r is the number of selected maximum eigenvalues.

[0022] Formula (1) is the expression of the failure time t, so the time-varying state function containing the random process can be expressed as G = g(z, t) after discretization of the random process. Where z is the design variable vector without time-varying factors, and t is the failure time.

[0023] In step two, the sampling based on the inverse reliability analysis principle is as follows:

[0024] The constraint of inverse reliability analysis is a simple spherical constraint, and sampling on the constraint surface is simple and easy to implement, greatly reducing the sampling range and improving the sampling efficiency. Sampling based on the inverse reliability analysis principle makes the selected sample points located on a circle or a hyper-spherical surface with β as the radius in the standard normal space, so as to approach the real MPP point as much as possible and ensure the solution accuracy of the response surface. Taking two-dimensional random variables as an example, the sampling in the standard normal space is shown in Figure 2 .

[0025] In principle, as long as the distance from the origin of the standard normal space meets the distance standard, the point with reliability index β can be used as a sample point, but in order to ensure the uniformity of sampling, the sample points are uniformly selected on the circle or hyper-sphere with β as the radius. Taking two-dimensional random variables as an example, if the number of primary sample points is m, the value of the sample point is:

[0026] θ=360° / m (2)

[0027] In order to facilitate calculation, θ=45° is taken, and thus a set of sample points of two random variables in the standard normal space can be obtained by using sampling based on the inverse reliability analysis principle. Finally, the sample points of the standard normal space are converted to the original space, and the conversion formula is as follows:

[0028] x=uσ+μ (3)

[0029] In the formula, u represents the variable in the standard normal space, x represents the variable in the original space, σ and μ are the mean square deviation and mean value of x respectively, and higher dimensional cases can be obtained by permutation and combination.

[0030] In step three, the extreme value of the limit state function of the transmission system facing the sampling data is specifically:

[0031] Based on the sampling data generated in step two, the design variables of the transmission system except the time-varying variables are brought into the limit state function to form a new function g(y(t),t) containing only time-varying variables. The extreme value of the new function is calculated by using a conventional optimization algorithm (such as particle swarm algorithm).

[0032] In step four, the extreme response surface of the limit state function of the transmission system based on the neural network is specifically:

[0033] A three-layer forward network neural network is used to construct the response surface, and the sample points obtained in step 2 are used as input signals. The extreme value of the limit state function is used as the output layer, and the sample data of the limit state function is trained. The unit layer receiving the input signal is called the input layer, the unit layer outputting the signal is called the output layer, and the unit layer not directly connected with the input and output is called the intermediate layer or the hidden layer. From the input layer to the hidden layer is a kind of fixed nonlinear transformation, which directly maps the input vector to a new space. The mapping from the hidden layer space to the output layer space is linear, and the output layer realizes linear weighted combination in the new linear space. The weight here is the adjustable parameter of the network.

[0034] The Euclidean distance between the test point and the sample point is taken as the independent variable, that is, it is assumed that represents a set of input vectors, is a basis function. Where ||x-x j || is the Euclidean distance:

[0035] (x-x j ) T (x-x j ), and 0.2≤c≤3. Ω represents a set of input vectors, denotes an N-dimensional real number field, and c is a norm factor.

[0036] In step five, the inverse reliability analysis model of the transmission system is constructed by using the extreme value response surface, and the specific process is as follows:

[0037] The constructed inverse reliability analysis model of the transmission system is as follows:

[0038]

[0039] In the formula, g(u) represents the extreme value response surface of the limit state function of the transmission system, β t represents a given reliability index. The optimal solution u * of the mathematical model (3) is also called the MPP point, and for the minimum value (probability performance measure) g(u * ) of the limit state function, if g(u * )≥0, it indicates that the reliability of the probability constraint is greater than or equal to the given reliability Φ(β t ), otherwise, the probability constraint does not meet the given reliability requirement.

[0040] In step six, the particle swarm algorithm is used to solve the constructed inverse reliability analysis model of the transmission system, and the specific process is as follows:

[0041] Step 1: initialize the particle swarm, each particle position, velocity and calculate its fitness function value, mark the current position of the particle as P i,best,0 , obtain the global optimal position G i,best,0 and its fitness function value, and the fitness function uses the objective function, that is, g(u) instead.

[0042] Step 2: set the iteration number n=1.

[0043] Step 3: if n max , turn to Step 4; otherwise, turn to Step 11, and N max represents the maximum iteration step length set.

[0044] Step 4: update the adaptive inertia weight w according to formula (4).

[0045]

[0046] In the formula, g i represents the global optimal position, x i is a position of the particle in the i direction in the D-dimensional space. X i,minX represents the worst particle in fitness, X i,max X represents the best particle in fitness.

[0047] Step 5: Update the velocity of each particle according to formula (5).

[0048] V i = ωV i + c1r1×(P besti -X i )+c2r2×(G besti -X i ) (6)

[0049] ω in formula (6) is a non-negative number, called inertia weight; c1 and c2 are learning factors; r1 and r2 are random numbers with a value range of [0, 1]; each particle has a speed limit [-Vmin, Vmax] and a position limit [Xmin, Xmax]. The first part of formula (6) is the memory term, which represents the previous speed and position of the particle; the second part is the individual cognition term, which represents that the movement of the particle comes from its own thinking; the third part is the social term, which represents the information sharing and group cooperation between particles; the three parts together determine how the particle moves next.

[0050] Step 6: Update the position of each particle according to formula (7).

[0051] X i = X i + V i (7)V i is the flight speed of the particle, X i is the current position of the particle.

[0052] Step 7: Calculate the fitness function value of each particle, and compare the calculated fitness function value to update the particle P i,best,n .

[0053] Step 8: After iteration, compare the current particle with the historical global best particle to update the global best G i,best,n .

[0054] Step 9: Output the best particle related information of each generation, including the value of each design variable of the best particle and the target function value.

[0055] Step 10: Set n = n + 1.

[0056] Step 11: Output the optimization result, i.e. the target function value of the optimal solution and the value of each design variable.

[0057] (III) Advantages and effects of the present application

[0058] The present application adopts a response surface method to approximately consider the extreme value of a limit state function of a transmission system time-varying factor, finds out the critical failure points of the state function at each time point through discretization of time according to the extreme value response principle, and thus converts the transmission system time-varying reliability analysis into a conventional reliability problem for solving. BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 is a general flowchart of the method of the present application.

[0060] Figure 2 is a sampling schematic diagram based on the PMA principle.

[0061] Figure 3 is a three-layer forward neural network structure.

[0062] Figure 4 is a process diagram of the neural network-based time-varying reliability analysis.

[0063] Figure 5 is a neural network response surface. DETAILED DESCRIPTION

[0064] The technical solutions of the present application will be described below in combination with specific examples.

[0065] The present application mainly adopts a response surface method to approximately consider the extreme value of a limit state function of a time-varying factor, finds out the critical failure points of the state function at each time point through discretization of time according to the extreme value response principle, and thus converts the time-varying reliability analysis into a conventional reliability problem for solving. The implementation steps will be described below with a gear transmission example.

[0066] I. Analysis of the gear transmission, fully considering the influences of manufacturing and processing, product assembly, material properties, applied load and external environment and other factors in the actual design process, since the present application mainly solves the reliability optimization analysis considering the time-varying uncertainty, the present application mainly considers two design variables of the fatigue degree X1 of the gear and the fatigue degree X2 of the bearing, and sets that X1 and X2 respectively obey normal distribution, and the mean and variance are respectively 3.5 and 0.09.

[0067] II. Set the time-varying limit state function G(X, t) as:

[0068]

[0069] Wherein, t represents time, when G(X, t) is less than 0, it is determined as failure, thus the failure probability P of the time-varying system f The time-varying function can be calculated by formula (9):

[0070]

[0071] In the formula, P f When T takes 1, 2, 3, 4 and 5 respectively, the time-varying reliability in the corresponding time period can be obtained.

[0072] Three, according to the sampling method of step two in the technical scheme, first specify The sampling area is shown in Figure 2 The number of initial sample points is 8, θ = 45°, and the initial sample point set is obtained by Latin hypercube sampling (3.77, 3.5), (3.69, 3.69), (3.5, 3.77), (3.31, 3.69), (3.23, 3.5), (3.31, 3.31), (3.5, 3.23), (3.69, 3.31).

[0073] Four, based on the initial sample point set, the limit state function value of each sample point is calculated by formula (8), and the response surface is constructed by using three-layer forward network neural network, and the sample points obtained in step 2 are used as input signals. The corresponding limit state function value is calculated.

[0074] Five, the inverse reliability analysis model constructed is solved by particle swarm algorithm, if convergence, end, otherwise continue iteration and update. The reliability analysis results are shown in the following table:

[0075]

[0076] N call Indicates the number of calls of particle swarm algorithm, P f Indicates the failure probability. It can be seen that with the expansion of the time period range, the failure probability P f First gradually increases and then remains unchanged, and the limit value of the time-varying function appears near t = 2, which is consistent with the calculation result. Figure 4 The limit value of the time-varying function appears near t = 2, which is consistent with the calculation result. Figure 5 The neural network response surface constructed for this example.

[0077] Based on the above analysis, when the fatigue degree X1 of the gear and the fatigue degree X2 of the bearing are 3.3184 and 3.2775 respectively, the limit state function G(X, t) is greater than 0, which meets the reliability requirement.

[0078] According to the above results, reasonable measures are taken to improve the reliability of gear transmission.

Claims

1. A time-varying reliability analysis method based on neural network extreme response surface, in particular to an aero-engine transmission system, characterized in that, The method comprises the following steps: Step 1: discretization of time-varying random process of a transmission system; Step 2: sampling based on inverse reliability analysis principle; Step 3: solving extreme value of a limit state function of the transmission system based on the sampling data; Step 4: constructing extreme value response surface of the limit state function of the transmission system based on a neural network; Step 5: constructing an inverse reliability analysis model of the transmission system by using the extreme value response surface; Step 6: solving the constructed inverse reliability analysis model of the transmission system by using a particle swarm algorithm. In step 1, the discretization of the time-varying random process of the transmission system is specifically as follows: For a limit state function g(x, y(t), t) of an aero-engine transmission system containing time-varying factors, all random processes are Gaussian random processes, and all random variables follow normal distribution; the optimal linear estimation expansion method is used to discretize the Gaussian random process; the specific implementation process is as follows: Let the mean and variance of the Gaussian random process y(t) be μ Y (t) and σ(t), the Gaussian random process is discretized using the optimal linear estimation expansion method. Let the random process be discretized into P time segments within a time period T, y(t) is expressed as: where ζ i are independent normal random variables; λ i and Φ i are the eigenvalues and eigenvectors of the correlation matrix C, where the eigenvalues are arranged in descending order, and the correlation matrix is expressed as C i,j = {p X (t i , t j ), i, j = 1, 2,..., P}, p X (t i , t j ) is the probability density from t i to t j , C i,j is the set of P time period probability densities within the time period T; and r is the number of the largest eigenvalues selected. Formula (1) is an expression of the failure time t, so the time-varying state function containing the random process is G=g(z, t) after the random process discretization; wherein z is a design variable vector not containing time-varying factors, and t is the failure time; In step 4, the construction of the extreme value response surface of the limit state function of the transmission system based on the neural network is specifically as follows: A three-layer forward network neural network is used to construct the response surface, the sample points obtained in step 2 are used as input signals, and the extreme value of the corresponding limit state function is used as the output layer, and the sample data of the limit state function are trained; the unit layer receiving the input signal is called the input layer, the unit layer outputting the signal is called the output layer, and the unit layer not directly connected with the input and output is called the intermediate layer or the hidden layer; the mapping from the input layer to the hidden layer is a fixed nonlinear transformation, which directly maps the input vector to a new space; the mapping from the hidden layer space to the output layer space is linear, and the output layer realizes linear weighted combination in the new linear space, wherein the weight is the adjustable parameter of the network; Let d(x, y) be the Euclidean distance between x and y, and let represent a set of input vectors, be a basis function; where ||x-x j || is the Euclidean distance: (x-x j ) T (x-x j ), and 0.2≤c≤3; Ω represents a set of input vectors, represents the N-dimensional real number field, and c is a norm factor; In step 5, the construction of the inverse reliability analysis model of the transmission system by using the extreme value response surface is specifically as follows: The constructed inverse reliability analysis model of the transmission system is as follows: where g(u) represents the extreme response surface of the limit state function of the transmission system, β t denotes the given reliability index; the optimal solution u* of formula (4) is also called the MPP point, and for the minimum value g(u*) of the limit state function, if g(u*)≥0, it indicates that the reliability of the probability constraint is greater than or equal to the given reliability Φ(β t ), otherwise, the probability constraint does not meet the given reliability requirement; In step 6, the solving of the constructed inverse reliability analysis model of the transmission system by using the particle swarm algorithm is specifically as follows: Step 1: initialize the particle swarm each particle position, velocity and calculate its fitness function value, mark the particle current position as P i,best,0 , get the global best position G i,best,0 and its fitness function value, fitness function using the objective function, that is, g(u) instead of; Step 2: set the iteration number n=1; Step 3: If n < N max , go to Step 4; otherwise, go to Step 11, N max represents the maximum iteration step set; Step 4: update the adaptive inertia weight w according to formula (5); where g i represents the global optimal position, x i is the current position of the particle in the i direction in the D-dimensional space; x i,min represents the worst particle, x i,max represents the best particle; Step 5: update the speed of each particle according to formula (6); V i = ωV i + c1r1x (P besti - X i ) + c2r2x (G besti - X i ) (6) ω in formula (6) is a non-negative number, called inertia weight; c1 and c2 are learning factors; r1 and r2 are random numbers with a value range of [0, 1]; each particle has a speed limit [-Vmin, Vmax] and a position limit [Xmin, Xmax]; the first part of formula (6) is the memory term, which represents the previous speed and position of the particle; the second part is the individual cognitive term, which represents that the action of the particle comes from its own thinking; the third part is the social term, which represents the information sharing and group cooperation between particles; the three parts jointly determine how the particle moves next time; Step 6: update the position of each particle according to formula (7); X i = X i + V i (7) V i is the flying speed of the particle, X i is the current position of the particle; Step 7: Calculate the fitness function value of each particle, and compare and update each particle P with the calculated fitness function value i,best,n ; Step 8: After iteration, the current particle is compared with the historical global best particle to update the global best G i,best,n ; Step 9: output the best particle related information of each generation, including the best particle of each design variable value and the objective function value; Step 10: set n = n + 1; Step 11: output the optimization results, that is, the objective function value of the optimal solution and the value of each design variable.

2. The time-varying reliability analysis method based on neural network extremal response surface according to claim 1, characterized in that: In step two, the sampling based on the inverse reliability analysis principle is as follows: The sampling based on the inverse reliability analysis principle is used, and as long as the distance from the origin of the standard normal space is the point of reliability index β, it can be used as a sample point, and the sample points are selected uniformly on the circle or hyper-spherical surface with β as the radius; if the number of initial sample points is m, the value of the sample point is: θ = 360° / m (2) θ = 45° is taken, and thus the sampling based on the inverse reliability analysis principle is used to obtain a set of sample points of two random variables in the standard normal space, and finally the sample points in the standard normal space are converted to the original space, that is, the conversion formula is as follows: x = uσ + μ (3) In the formula, u represents the variable in the standard normal space, x represents the variable in the original space, σ and μ are the mean square deviation and mean value of x respectively.

3. The time-varying reliability analysis method based on neural network extremal response surface of claim 1, wherein: In step three, the extreme value of the limit state function of the transmission system is solved based on the sampling data, which is as follows: Based on the sampling data generated in step two, the design variables of the transmission system except the time-varying variables are brought into the limit state function to form a new function g(y(t),t) containing only time-varying variables, and the optimization algorithm is used to calculate the extreme value of the new function.

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