An improved ABC-Kriging method for reliability optimization design of mechanical systems

Abstract

The present invention is a mechanical system reliability optimization design method of improving ABC-Kriging, including setting optimization design variables and interval ranges, obtaining initial sample points, simulating to obtain true response values, selecting relevant functions of the fitting data sample library, and using the improved ABC algorithm to obtain the optimal parameter values, establishing an improved ABC-Kriging proxy model, and calculating the reliability, judging whether the reliability meets the accuracy requirements, if so, establishing a reliability optimization design model, if not, evaluating the sampled data points, reselecting samples and refitting, repeating the above steps, and then solving the reliability optimization design model. The present invention improves the calculation accuracy and efficiency of the traditional ABC algorithm, obtains the optimal kernel function parameters and then constructs the improved ABC-Kriging proxy model, improves the construction accuracy of the proxy model, proposes the B criterion and H criterion sample point judgment functions of parallel addition points, and further improves the accuracy and calculation efficiency of the mechanical system reliability optimization design.

Description

Technical Field

[0001] The invention relates to the technical field of structural optimization design, and in particular to an improved ABC-Kriging mechanical system reliability optimization design method. Background Art

[0002] The purpose of mechanical system reliability optimization design is to maximize the system performance to meet the requirements of reliability indicators under the influence of uncertain design parameters, and the accuracy and efficiency of its optimization design are the key to improving system performance. Mechanical systems have complex structures and many parts, and there are problems such as difficulty in obtaining physical models and multiple optimization variables. Especially for simple parts that play a key role, the accuracy and reliability of their optimization results should be improved; however, traditional optimization design methods are mostly based on the physical model of the structural system. Although the accuracy of the optimization results can be guaranteed, the calculation efficiency is low and it is not applicable to complex systems; although the optimization design based on the proxy model guarantees the optimization efficiency of the structural system, its optimization results are greatly affected by the accuracy of the proxy model and the efficiency of the intelligent algorithm, and its universality is poor. Summary of the invention

[0003] In order to solve the technical problems in the prior art of mechanical system optimization design that the physical model is difficult to obtain, the traditional ABC algorithm is difficult to calculate, and the calculation efficiency is low, the present invention provides a mechanical system reliability optimization design method based on an improved ABC-Kriging.

[0004] The technical solution adopted by the present invention to achieve the above-mentioned purpose is: an improved ABC-Kriging mechanical system reliability optimization design method, comprising the following steps:

[0005] S1: Set the mechanical system optimization design variable x and the allowed range of each optimization design variable;

[0006] S2: Use the Latin hypercube method to sample the optimization design variable x to obtain the initial sample points of the optimization design variable;

[0007] S3: Use the Matlab and ANSYS joint simulation platform to collaboratively calculate the true response value y of the mechanical system corresponding to each initial sample point in S2, and classify the response set into the mechanical system data sample library;

[0008] S4: Select the correlation function of the fitting data sample library and use the improved ABC algorithm to obtain the optimal θ value;

[0009] S5: Establish an improved ABC-Kriging proxy model and use importance sampling techniques to calculate reliability;

[0010] S6: If the calculation accuracy of the reliability meets the requirements, the mechanical system reliability optimization design model is established by combining the improved ABC-Kriging proxy model and the allowed range of the optimization design variables; if the calculation accuracy of the reliability does not meet the requirements, the importance sampling data points are evaluated using the adaptive B criterion and H criterion, and the data points that meet the requirements and their corresponding response values ​​are added to the sample library, and the Euclidean distance is used to delete duplicate sample points, and S4-S6 is repeated;

[0011] S7: Use the improved ABC algorithm to solve the reliability optimization design model and obtain the optimal value of the optimization variable.

[0012] Preferably, in step S1, the mechanical system optimization design variables are set as: x = (x1, x2, ..., x n ), the allowable range of each optimization design variable is:

[0013] Preferably, the initial sample points for optimizing the design variables in step S2 are:

[0014] Preferably, the true response value in step S3 is: y=(y1, y2, ..., y n );

[0015] Preferably, step S4 comprises the following steps:

[0016] S4.1: Select the Kriging surrogate model as the function to fit the database samples and optimize the design variables x = (x1, x2, ..., x n ) and its corresponding true response value y=(y1,y2,...,y n ) is:

[0017]

[0018] in: is the response value after fitting, f(·) is the polynomial function of the optimized design variable x, β is the coefficient of the polynomial function, z(x) is a random statistical process, and the covariance function in the random statistical process is:

[0019]

[0020] Where: Cov(·,·) is the covariance function, z(x i )、z(x j ) are sample points x i and x j A random statistical process, is the variance, R ij (θ,x i ,xj ) is the sample point x i and x j The spatial correlation kernel function of , θ is the parameter of the spatial correlation kernel function, i and j are the sample numbers;

[0021] S4.2: Use the Gaussian kernel function as the spatial correlation kernel function, and obtain the optimal θ value by solving the minimization problem. The expression of the correlation kernel function is:

[0022]

[0023] in: is the probability density function under the standard distribution, R(θ) is the kernel function with parameter θ, m is the number of optimized design variables, σ 2 is the variance;

[0024] S4.3: Solve S4.2 using the improved ABC algorithm to obtain the optimal θ value, including the following steps:

[0025] S4.3.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0026] x i =x min,i +rand×(x max,i -x min,i )

[0027] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0028] S4.3.2: Substitute the minimization problem in S4.2 into the fitness function as the objective and calculate the fitness, which is expressed as:

[0029]

[0030] Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0031] S4.3.3: Record the current optimal fitness value

[0032] S4.3.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0033]

[0034] Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0035]

[0036] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0037] S4.3.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0038]

[0039] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0040] S4.3.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, stop updating the nectar source location and output the optimal θ value based on the current optimal fitness. If trail < N, continue to update the nectar source location. The expression is:

[0041]

[0042] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location;

[0043] Preferably, the expression of the improved ABC-Kriging surrogate model is:

[0044]

[0045] in: represents the mechanical system performance function proxy model, x i represents the i-th sample point, f(·) is the polynomial function, β is the coefficient of the polynomial function, r is the correlation vector between the prediction point and the sample point, R is the spatial correlation kernel function, θ is the parameter of the spatial correlation kernel function, y is the true response value, and J is the sample point regression matrix;

[0046] The reliability of the above formula is calculated using the important sampling technique, and the expression is:

[0047]

[0048] Where: R represents reliability, P f is the failure probability, E represents the expectation, I represents the indicative function, represents the proxy model of mechanical system performance function, x represents the sample point, g(x) is the probability density function of the performance function, p v (x) is the importance sampling density;

[0049] Preferably, the reliability calculation accuracy in step S6 satisfies the required expression:

[0050]

[0051] in: is the reliability value of the kth sample point, is the reliability value at the k-1th sample point, ε=0.01;

[0052] If the reliability meets the accuracy requirements, the mechanical system reliability optimization design model is established, and its expression is:

[0053]

[0054] stPr{y i (d,x,C)≥0}≥[R i ],i=1,2,...,m

[0055] h j (x)≤0,j=1,2,...,n

[0056] G k (x)=0,k=1,2,...,l

[0057] x L ≤x≤x U ;

[0058] Where: f(·) is the objective function, d is the deterministic design variable, x is the optimization design variable, C is the random parameter; μ x is the mean of the optimized design variables; Pr(·) is the probability of the structural performance function; y i represents the predicted performance function; [R i ] is the reliability required for structural design, h j (x) and G k(x) are nonlinear equality and inequality constraints, respectively. U and x L are the upper and lower limits of the optimization design variables, respectively. and are the upper and lower limits of the mean of the optimized design variables, respectively; i, j, k represent the corresponding serial numbers, and m, n, l represent the corresponding quantities, respectively;

[0059] Preferably, the expression of the adaptive B criterion in step S6 is:

[0060]

[0061] The H criterion expression is:

[0062]

[0063] Among them: B(x) is the B criterion, H(x) is the H criterion, is the mean of the response values ​​after fitting, is the variance of the response value after fitting, x and is the sample point and its corresponding fitted response value, Φ is the cumulative distribution function, and φ is the probability density function;

[0064] Preferably, in step S7, solving the reliability optimization design model using the improved ABC algorithm is specifically as follows:

[0065] S7.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0066] x i =x min,i +rand×(x max,i -x min,i )

[0067] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0068] S7.2: Substitute the objective function in the mechanical system reliability optimization design model into the fitness function as the target and calculate the fitness. The fitness function expression is:

[0069]

[0070] Among them: fit i is the fitness value, f iis the objective function, i represents the sample number, and n represents the number of samples;

[0071] S7.3: Record the current optimal fitness value

[0072] S7.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0073]

[0074] Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0075]

[0076] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0077] S7.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0078]

[0079] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0080] S7.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, then stop updating the nectar source location, and output the optimization result of the optimal optimization design variable according to the current optimal fitness. If trail < N, then continue to update the nectar source location. The expression is:

[0081]

[0082] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location.

[0083] The present invention improves the computational accuracy and efficiency of the traditional ABC algorithm by introducing an adaptive step size and a new roulette strategy, and uses them to obtain the optimal kernel function parameters to construct an improved ABC-Kriging proxy model, thereby improving the construction accuracy of the proxy model; at the same time, a parallel-pointed B criterion and H criterion sample point judgment function are proposed, and a reliability optimization design model based on the improved ABC-Kriging proxy model is constructed accordingly. By solving the improved ABC algorithm, the accuracy and computational efficiency of the reliability optimization design of the mechanical system are further improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 It is a flow chart of the method of the present invention.

[0085] Figure 2 It is a comparison diagram of the improved ABC-Kriging method and the traditional method in the second embodiment of the present invention.

[0086] Figure 3 This is the improved ABC-Kriging method fitting residual graph in the second embodiment of the present invention.

[0087] Figure 4 It is a reliability optimization design iteration diagram in the second embodiment of the present invention.

[0088] Figure 5 This is a mechanical structure diagram of a ten-bar truss in Embodiment 3 of the present invention.

[0089] Figure 6 This is the residual graph of the improved ABC-Kriging method fitting in the third embodiment of the present invention.

[0090] Figure 7 It is a reliability optimization design iteration diagram in the third embodiment of the present invention. DETAILED DESCRIPTION

[0091] The specific implementation of the present invention is as follows:

[0092] Embodiment 1: A mechanical system reliability optimization design method based on improved ABC-Kriging, comprising the following steps:

[0093] The following steps are involved:

[0094] S1: Set the mechanical system optimization design variable x and the allowed range of each optimization design variable. The mechanical system optimization design variable is: x = (x1, x2, ..., x n ), the allowable range of each optimization design variable is:

[0095]

[0096] S2: Use the Latin hypercube method to sample the optimization design variable x to obtain the initial sample points of the optimization design variable. The initial sample points of the optimization design variable are:

[0097] S3: Use Matlab and ANSYS joint simulation platform to collaboratively calculate the true response value y of the mechanical system corresponding to each initial sample point in S2. The true response value is: y=(y1,y2,...,y n ), and classify the response set into a mechanical system data sample library;

[0098] S4: Select the correlation function of the fitting data sample library and use the improved ABC algorithm to obtain the optimal θ value;

[0099] S4.1: Select the Kriging surrogate model as the function to fit the database samples and optimize the design variables x = (x1, x2, ..., x n ) and its corresponding true response value y=(y1,y2,...,y n ) is:

[0100]

[0101] in: is the response value after fitting, f(·) is the polynomial function of the optimized design variable x, β is the coefficient of the polynomial function, z(x) is a random statistical process, and the covariance function in the random statistical process is:

[0102]

[0103] Where: Cov(·,·) is the covariance function, z(x i )、z(x j ) are sample points x i and x j A random statistical process, is the variance, R ij (θ,x i ,x j ) is the sample point x i and x j The spatial correlation kernel function of , θ is the parameter of the spatial correlation kernel function, i and j are the sample numbers;

[0104] S4.2: Use the Gaussian kernel function as the spatial correlation kernel function, and obtain the optimal θ value by solving the minimization problem. The expression of the correlation kernel function is:

[0105]

[0106] in: is the probability density function under the standard distribution, R(θ) is the kernel function with parameter θ, m is the number of optimized design variables, σ 2 is the variance;

[0107] S4.3: Solve S4.2 using the improved ABC algorithm to obtain the optimal θ value, including the following steps:

[0108] S4.3.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0109] x i =x min,i +rand×(x max,i -x min,i )

[0110] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0111] S4.3.2: Substitute the minimization problem in S4.2 as the goal into the fitness function and calculate the fitness, that is, the quality of the nectar source in the algorithm, which is expressed as:

[0112]

[0113] Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0114] S4.3.3: Record the current optimal fitness value

[0115] S4.3.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0116]

[0117] Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0118]

[0119] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0120] S4.3.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0121]

[0122] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0123] S4.3.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, stop updating the nectar source location and output the optimal θ value based on the current optimal fitness. If trail < N, continue to update the nectar source location. The expression is:

[0124]

[0125] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location;

[0126] S5: Establish an improved ABC-Kriging proxy model and use importance sampling techniques to calculate reliability;

[0127] The expression of the improved ABC-Kriging surrogate model is:

[0128]

[0129] in: represents the mechanical system performance function proxy model, x i represents the i-th sample point, f(·) is the polynomial function, β is the coefficient of the polynomial function, r is the correlation vector between the prediction point and the sample point, R is the spatial correlation kernel function, θ is the parameter of the spatial correlation kernel function, y is the true response value, and J is the sample point regression matrix;

[0130] The reliability of the above formula is calculated using the important sampling technique, and the expression is:

[0131]

[0132] Where: R represents reliability, P f is the failure probability, E represents the expectation, I represents the indicative function, represents the proxy model of mechanical system performance function, x represents the sample point, g(x) is the probability density function of the performance function, p v (x) is the importance sampling density

[0133] S6: If the calculation accuracy of the reliability meets the requirements, the mechanical system reliability optimization design model is established by combining the improved ABC-Kriging proxy model and the allowed range of the optimization design variables; if the calculation accuracy of the reliability does not meet the requirements, the importance sampling data points are evaluated using the adaptive B criterion and H criterion, and the data points that meet the requirements and their corresponding response values ​​are added to the sample library, and the Euclidean distance is used to delete duplicate sample points, and S4-S6 is repeated;

[0134] The expression that meets the requirements of reliability calculation accuracy is:

[0135]

[0136] in: is the reliability value of the kth sample point, is the reliability value at the k-1th sample point, ε=0.01;

[0137] If the reliability meets the accuracy requirements, the mechanical system reliability optimization design model is established, and its expression is:

[0138]

[0139]

[0140] h j (x)≤0,j=1,2,...,n

[0141] G k (x)=0,k=1,2,...,l

[0142] x L ≤x≤x U ;

[0143] Where: f(·) is the objective function, d is the deterministic design variable, x is the optimization design variable, C is the random parameter; μ x is the mean of the optimized design variables; Pr(·) is the probability of the structural performance function; y i represents the predicted performance function; [R i ] is the reliability required for structural design, h j (x) and G k (x) are nonlinear equality and inequality constraints, respectively. U and x Lare the upper and lower limits of the optimization design variables, respectively. and are the upper and lower limits of the mean of the optimized design variables, respectively; i, j, k represent the corresponding serial numbers, and m, n, l represent the corresponding quantities, respectively;

[0144] The expression of the adaptive B criterion is:

[0145]

[0146] The H criterion expression is:

[0147]

[0148] Among them: B(x) is the B criterion, H(x) is the H criterion, is the mean of the response values ​​after fitting, is the variance of the response value after fitting, x and is the sample point and its corresponding fitted response value, Φ is the cumulative distribution function, and φ is the probability density function;

[0149] S7: The reliability optimization design model is solved using the improved ABC algorithm as follows:

[0150] S7.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0151] x i =x min,i +rand×(x max,i -x min,i )

[0152] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0153] S7.2: Substitute the objective function in the mechanical system reliability optimization design model into the fitness function as the target and calculate the fitness. The fitness function is expressed as:

[0154]

[0155] Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0156] S7.3: Record the current optimal fitness value

[0157] S7.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0158]

[0159] Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0160]

[0161] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0162] S7.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0163]

[0164] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0165] S7.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, then stop updating the nectar source location, and output the optimization result of the optimal optimization design variable according to the current optimal fitness. If trail < N, then continue to update the nectar source location. The expression is:

[0166]

[0167] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location.

[0168] Embodiment 2: A mechanical system reliability optimization design method based on improved ABC-Kriging, comprising the following steps:

[0169] S1: Set the optimization design variables of the mechanical system to x1 and x2 respectively, and the limit state equation of the mechanical system is:

[0170] y=-x1sin(4x1)-1.1x2sin(2x2),

[0171] All optimization design variables obey normal distribution, among which x1~N(2.97,0.1 2 ), x2~N(3.4,0.1 2 ), thus the allowed range of each optimization design variable is x1∈[0,3.7], x2∈[0,4];

[0172] S2: Use the Latin hypercube method to sample the optimization design variable x to obtain the initial sample points of the optimization variable. The initial sample points of the optimization design variable are:

[0173] S3: Use Matlab and ANSYS joint simulation platform to collaboratively calculate the true response value y of the mechanical system corresponding to each initial sample point in S2. The true response value is: y=(y1,y2,...,y n ), and classify the response set into a mechanical system data sample library;

[0174] S4: Select the correlation function of the fitting data sample library and use the improved ABC algorithm to obtain the optimal θ value;

[0175] S4.1: Select the Kriging surrogate model as the function to fit the database samples, and optimize the design variables x = (x1, x2) and their corresponding true response values ​​y = (y1, y2, ..., y n ) is:

[0176]

[0177] in: is the response value after fitting, f(·) is the polynomial function of the optimized design variable x, β is the coefficient of the polynomial function, z(x) is a random statistical process, and the covariance function in the random statistical process is:

[0178]

[0179] Where: Cov(·,·) is the covariance function, z(x i )、z(x j ) are sample points x i and x j A random statistical process, is the variance, R ij (θ,x i ,x j ) is the sample point x i and x j The spatial correlation kernel function of , θ is the parameter of the spatial correlation kernel function, i and j are the sample numbers;

[0180] S4.2: Use the Gaussian kernel function as the spatial correlation kernel function, and obtain the optimal θ value by solving the minimization problem. The expression of the correlation kernel function is:

[0181]

[0182] in: is the probability density function under the standard distribution, R(θ) is the kernel function with parameter θ, m is the number of optimized design variables, σ 2 is the variance;

[0183] S4.3: Solve S4.2 using the improved ABC algorithm to obtain the optimal θ value, including the following steps:

[0184] S4.3.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0185] x i =x min,i +rand×(x max,i -x min,i )

[0186] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0187] S4.3.2: Substitute the minimization problem in S4.2 as the goal into the fitness function and calculate the fitness, that is, the quality of the nectar source in the algorithm, which is expressed as:

[0188]

[0189] Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0190] S4.3.3: Record the current optimal fitness value

[0191] S4.3.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0192]

[0193] Where: v iis the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0194]

[0195] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0196] S4.3.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0197]

[0198] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0199] S4.3.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, stop updating the nectar source location and output the optimal θ value based on the current optimal fitness. If trail < N, continue to update the nectar source location. The expression is:

[0200]

[0201] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location;

[0202] S5: Establish an improved ABC-Kriging surrogate model. The fitting graph and residual of the surrogate model are as follows Figure 2 and Figure 3 As shown by Figure 2 It can be seen that the fitting curve of this method has a higher degree of coincidence with the true limit state curve, and has a better fitting effect than the traditional method. The importance sampling technique is then used to calculate the reliability;

[0203] The expression of the improved ABC-Kriging surrogate model is:

[0204]

[0205] in: represents the mechanical system performance function proxy model, xi represents the i-th sample point, f(·) is the polynomial function, β is the coefficient of the polynomial function, r is the correlation vector between the prediction point and the sample point, R is the spatial correlation kernel function, θ is the parameter of the spatial correlation kernel function, y is the true response value, and J is the sample point regression matrix;

[0206] The reliability of the above formula is calculated using the important sampling technique, and the expression is:

[0207]

[0208] Where: R represents reliability, P f is the failure probability, E represents the expectation, I represents the indicative function, represents the proxy model of mechanical system performance function, x represents the sample point, g(x) is the probability density function of the performance function, p v (x) is the importance sampling density

[0209] S6: If the calculation accuracy of the reliability meets the requirements, the mechanical system reliability optimization design model is established by combining the improved ABC-Kriging proxy model and the allowed range of the optimization design variables; if the calculation accuracy of the reliability does not meet the requirements, the importance sampling data points are evaluated using the adaptive B criterion and H criterion, and the data points that meet the requirements and their corresponding response values ​​are added to the sample library, and the Euclidean distance is used to delete duplicate sample points, and S4-S6 is repeated;

[0210] The expression that meets the requirements of reliability calculation accuracy is:

[0211]

[0212] in: is the reliability value of the kth sample point, is the reliability value at the k-1th sample point, ε=0.01;

[0213] If the reliability meets the accuracy requirements, the mechanical system reliability optimization design model is established, and its expression is:

[0214]

[0215] stβ≥2.0

[0216] 0≤x1≤3.7; 0≤x2≤4

[0217] Where: f(·) is the objective function, x1 and x2 are optimization design variables, and β is the coefficient of the polynomial function;

[0218] The expression of the adaptive B criterion is:

[0219]

[0220] The H criterion expression is:

[0221]

[0222] Among them: B(x) is the B criterion, H(x) is the H criterion, is the mean of the response values ​​after fitting, is the variance of the response value after fitting, x and is the sample point and its corresponding fitted response value, Φ is the cumulative distribution function, and φ is the probability density function;

[0223] S7: In step S7, the reliability optimization design model is solved by using the improved ABC algorithm as follows:

[0224] S7.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0225] x i =x min,i +rand×(x max,i -x min,i )

[0226] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0227] S7.2: Substitute the objective function in the mechanical system reliability optimization design model into the fitness function as the target and calculate the fitness. The fitness function expression is:

[0228]

[0229] Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0230] S7.3: Record the current optimal fitness value

[0231] S7.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0232]

[0233] Where: v iis the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0234]

[0235] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0236] S7.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0237]

[0238] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0239] S7.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, then stop updating the nectar source location, and output the optimization result of the optimal optimization design variable according to the current optimal fitness. If trail < N, then continue to update the nectar source location. The expression is:

[0240]

[0241] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location. The optimization results of the optimization design variables are as follows: Figure 4 shown.

[0242] Table 1 gives a comparison of the reliability optimization design results of Example 2. Figure 4 As shown in Table 1, the objective function value in method 1 tends to be stable earlier than that in method 2. This shows that this method has a faster calculation speed and more accurate results.

[0243] Table 1 Comparison of reliability optimization design results of Example 2

[0244]

[0245] Embodiment three:

[0246] S1: A mechanical system can be expressed as Figure 5The ten-bar truss structure shown in the figure: The optimal design variables of the mechanical system are set to be A1, A2, and A3, and A1, A2, and A3 obey the normal distribution respectively: A1~N(13,0.1 2 )、A2~N(2,0.1 2 )、A3~N(9,0.1 2 ), in order to make the stiffness of the system meet the use requirements, the limit state is defined as the displacement of node 3 in the vertical direction does not exceed the set displacement, the expression is: y = d-u3(A), where d is the set displacement, d = 4.5, u3(A) is the displacement of node 3 in the vertical direction, that is, y must be greater than zero, u3(A) can be expressed as:

[0247]

[0248] in: Where P = 10 5 ,L=360,E=10 7 , from this, the allowable range of each optimization design variable is: A1∈[12.5,13.5], A2∈[1.8,2.2], A3∈[8.5,9.5];

[0249] S2: Use the Latin hypercube method to sample the optimization design variable x to obtain the initial sample points of the optimization variable. The initial sample points of the optimization design variable are:

[0250] S3: Use Matlab and ANSYS joint simulation platform to collaboratively calculate the true response value y of the mechanical system corresponding to each initial sample point in S2. The true response value is: y=(y1,y2,...,y n ), and classify the response set into a mechanical system data sample library;

[0251] S4: Select the correlation function of the fitting data sample library and use the improved ABC algorithm to obtain the optimal θ value;

[0252] S4.1: Select the Kriging surrogate model as the function to fit the database samples, and optimize the design variables A = (A1, A2, A3) and their corresponding true response values ​​y = (y1, y2, ..., y n ) is:

[0253]

[0254] in:: is the response value after fitting, f(·) is the polynomial function of the optimized design variable x, β is the coefficient of the polynomial function, z(x) is a random statistical process, and the covariance function in the random statistical process is:

[0255]

[0256] Where: Cov(·,·) is the covariance function, z(x i )、z(x j ) are sample points x i and x j A random statistical process, is the variance, R ij (θ,x i ,x j ) is the sample point x i and x j The spatial correlation kernel function of , θ is the parameter of the spatial correlation kernel function, i and j are the sample numbers;

[0257] S4.2: Use the Gaussian kernel function as the spatial correlation kernel function, and obtain the optimal θ value by solving the minimization problem. The expression of the correlation kernel function is:

[0258]

[0259] in: is the probability density function under the standard distribution, R(θ) is the kernel function with parameter θ, m is the number of optimized design variables, σ 2 is the variance;

[0260] S4.3: Solve S4.2 using the improved ABC algorithm to obtain the optimal θ value, including the following steps:

[0261] S4.3.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0262] x i =x min,i +rand×(x max,i -x min,i )

[0263] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0264] S4.3.2: Substitute the minimization problem in S4.2 as the goal into the fitness function and calculate the fitness, that is, the quality of the nectar source in the algorithm, which is expressed as:

[0265]

[0266] Among them: fit iis the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0267] S4.3.3: Record the current optimal fitness value

[0268] S4.3.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0269]

[0270] Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0271]

[0272] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0273] S4.3.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0274]

[0275] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0276] S4.3.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, stop updating the nectar source location and output the optimal θ value based on the current optimal fitness. If trail < N, continue to update the nectar source location. The expression is:

[0277]

[0278] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location;

[0279] S5: Establish an improved ABC-Kriging surrogate model, whose residual is as follows Figure 6 As shown, the reliability calculation is performed using the importance sampling technique;

[0280] The expression of the improved ABC-Kriging surrogate model is:

[0281]

[0282] in: represents the mechanical system performance function proxy model, x i represents the i-th sample point, f(·) is the polynomial function, β is the coefficient of the polynomial function, r is the correlation vector between the prediction point and the sample point, R is the spatial correlation kernel function, θ is the parameter of the spatial correlation kernel function, y is the true response value, and J is the sample point regression matrix;

[0283] The reliability of the above formula is calculated using the important sampling technique, and the expression is:

[0284]

[0285] Where: R represents reliability, P f is the failure probability, E represents the expectation, I represents the indicative function, represents the proxy model of mechanical system performance function, x represents the sample point, g(x) is the probability density function of the performance function, p v (x) is the importance sampling density

[0286] S6: If the calculation accuracy of the reliability meets the requirements, the mechanical system reliability optimization design model is established by combining the improved ABC-Kriging proxy model and the allowed range of the optimization design variables; if the calculation accuracy of the reliability does not meet the requirements, the importance sampling data points are evaluated using the adaptive B criterion and H criterion, and the data points that meet the requirements and their corresponding response values ​​are added to the sample library, and the Euclidean distance is used to delete duplicate sample points, and S4-S6 is repeated;

[0287] The expression that meets the requirements of reliability calculation accuracy is:

[0288]

[0289] in: is the reliability value of the kth sample point, is the reliability value at the k-1th sample point, ε=0.01;

[0290] If the reliability meets the accuracy requirements, the mechanical system reliability optimization design model is established, and its expression is:

[0291]

[0292] stβ≥2.4210

[0293] 12.5≤A1≤13.5, 1.8≤A2≤2.2, 8.5≤A3≤9.5

[0294] Where: f is the objective function, A1, A2, A3 are optimization design variables, and β is the coefficient of the polynomial function;

[0295] The expression of the adaptive B criterion is:

[0296]

[0297] The H criterion expression is:

[0298]

[0299] Among them: B(x) is the B criterion, H(x) is the H criterion, is the mean of the response values ​​after fitting, is the variance of the response value after fitting, x and is the sample point and its corresponding fitted response value, Φ is the cumulative distribution function, and φ is the probability density function;

[0300] S7: In step S7, the reliability optimization design model is solved by using the improved ABC algorithm as follows:

[0301] S7.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is:

[0302] x i =x min,i +rand×(x max,i -x min,i )

[0303] Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1);

[0304] S7.2: Substitute the objective function in the mechanical system reliability optimization design model into the fitness function as the target and calculate the fitness. The fitness function expression is:

[0305]

[0306] Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples;

[0307] S7.3: Record the current optimal fitness value

[0308] S7.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is:

[0309]

[0310] Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is:

[0311]

[0312] Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm;

[0313] S7.5: Use the improved roulette method to compare the optimization results, and the expression is:

[0314]

[0315] Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness,

[0316] S7.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, then stop updating the nectar source location, and output the optimization result of the optimal optimization design variable according to the current optimal fitness. If trail < N, then continue to update the nectar source location. The expression is:

[0317]

[0318] Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimal design variable corresponding to the current nectar source location, and the optimization results of the optimal design variables are as follows: Figure 7 shown.

[0319] Table 2 gives a comparison of the reliability optimization design results of Example 3. Figure 7As shown, the optimization target of this method tends to be stable earlier than the traditional method as the number of iterations increases. It can be seen that compared with the traditional method, this method has higher calculation efficiency and more accurate calculation results.

[0320] Table 2 Comparison of reliability optimization design results of Example 3

[0321]

[0322] The present invention is described by way of embodiments, and those skilled in the art will appreciate that various changes or equivalent substitutions may be made to these features and embodiments without departing from the spirit and scope of the present invention. In addition, under the teachings of the present invention, these features and embodiments may be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application belong to the protection scope of the present invention.

Claims

1. An improved ABC-Kriging method for mechanical system reliability optimization design, characterized in that: The following steps are involved: S1: Set the mechanical system optimization design variable x and the allowed range of each optimization design variable; S2: Use the Latin hypercube method to sample the optimization design variable x to obtain the initial sample points of the optimization design variable; S3: Use the Matlab and ANSYS joint simulation platform to collaboratively calculate the true response value y of the mechanical system corresponding to each initial sample point in S2, and classify the response set into the mechanical system data sample library; S4: Select the correlation function of the fitting data sample library and use the improved ABC algorithm to obtain the optimal θ value; S5: Establish an improved ABC-Kriging proxy model and use importance sampling techniques to calculate reliability; The expression of the improved ABC-Kriging surrogate model is: in: represents the mechanical system performance function proxy model, x i represents the i-th sample point, f(·) is the polynomial function, β is the coefficient of the polynomial function, r is the correlation vector between the prediction point and the sample point, R is the spatial correlation kernel function, θ is the parameter of the spatial correlation kernel function, y is the true response value, and J is the sample point regression matrix; The reliability of the above formula is calculated using the important sampling technique, and the expression is: Where: R represents reliability, P f is the failure probability, E represents the expectation, I represents the indicative function, represents the proxy model of the mechanical system performance function, x represents the sample point, g(x) is the probability density function of the performance function, p v (x) is the importance sampling density; S6: If the calculation accuracy of the reliability meets the requirements, the mechanical system reliability optimization design model is established by combining the improved ABC-Kriging proxy model and the allowed range of the optimization design variables; if the calculation accuracy of the reliability does not meet the requirements, the importance sampling data points are evaluated using the adaptive B criterion and H criterion, and the data points that meet the requirements and their corresponding response values ​​are added to the sample library, and the Euclidean distance is used to delete duplicate sample points, and S4-S6 is repeated; S7: Use the improved ABC algorithm to solve the reliability optimization design model and obtain the optimal value of the optimization variable.

2. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 1 is characterized in that: In step S1, the mechanical system optimization design variables are set as: x = (x1, x2, ..., x n ), the allowable range of each optimization design variable is:

3. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 2 is characterized in that: The initial sample points of the optimized design variables in step S2 are:

4. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 3 is characterized in that: The true response value in step S3 is: y=(y1,y2,...,y n ).

5. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 4 is characterized in that: Step S4 includes the following steps: S4.1: Select the Kriging surrogate model as the function to fit the database samples and optimize the design variables x = (x1, x2, ..., x n ) and its corresponding true response value y=(y1,y2,...,y n ) is: in: is the response value after fitting, f(·) is the polynomial function of the optimized design variable x, β is the coefficient of the polynomial function, z(x) is a random statistical process, and the covariance function in the random statistical process is: Where: Cov(·,·) is the covariance function, z(x i )、z(x j ) are sample points x i and x j A random statistical process, is the variance, R ij (θ,x i ,x j ) is the sample point x i and x j The spatial correlation kernel function of , θ is the parameter of the spatial correlation kernel function, i and j are the sample numbers; S4.2: Use the Gaussian kernel function as the spatial correlation kernel function, and obtain the optimal θ value by solving the minimization problem. The expression of the correlation kernel function is: in: is the probability density function under the standard distribution, R(θ) is the kernel function with parameter θ, m is the number of optimized design variables, σ 2 is the variance; S4.3: Solve S4.2 using the improved ABC algorithm to obtain the optimal θ value, including the following steps: S4.3.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is: x i =x min,i +rand×(x max,i -x min,i ) Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1); S4.3.2: Substitute the minimization problem in S4.2 into the fitness function as the objective and calculate the fitness, which is expressed as: Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples; S4.3.3: Record the current optimal fitness value S4.3.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is: Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is: Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm; S4.3.5: Use the improved roulette method to compare the optimization results, and the expression is: Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness, S4.3.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, stop updating the nectar source location and output the optimal θ value based on the current optimal fitness. If trail < N, continue to update the nectar source location. The expression is: Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location.

6. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 1 is characterized in that: The reliability calculation accuracy in step S6 meets the requirement as follows: in: is the reliability value of the kth sample point, is the reliability value at the k-1th sample point, ε=0.01; If the reliability meets the accuracy requirements, the mechanical system reliability optimization design model is established, and its expression is: stPr{y i (d,x,C)≥0}≥[R i ],i=1,2,...,m h j (x)≤0,j=1,2,...,n G k (x)=0,k=1,2,...,l Where: f(·) is the objective function, d is the deterministic design variable, x is the optimization design variable, C is the random parameter; μ x is the mean of the optimized design variables; Pr(·) is the probability of the structural performance function; y i represents the predicted performance function; [R i ] is the reliability required for structural design, h j (x) and G k (x) are nonlinear equality and inequality constraints, respectively. U and x L are the upper and lower limits of the optimization design variables, respectively. and are the upper and lower limits of the mean of the optimized design variables; i, j, k represent the corresponding serial numbers, and m, n, l represent the corresponding quantities.

7. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 6 is characterized in that: The expression of the adaptive B criterion in step S6 is: The H criterion expression is: Among them: B(x) is the B criterion, H(x) is the H criterion, is the mean of the response values ​​after fitting, is the variance of the response value after fitting, x and is the sample point and its corresponding fitted response value, Φ is the cumulative distribution function, and φ is the probability density function.

8. The improved ABC-Kriging mechanical system reliability optimization design method according to claim 7 is characterized in that: In step S7, the reliability optimization design model is solved by using the improved ABC algorithm as follows: S7.1: Initialize the number of nectar sources and flight search range in the ABC algorithm, and set the maximum number of optimization times N. The initialization formula is: x i =x min,i +rand×(x max,i -x min,i ) Where: x i is the optimal design variable value corresponding to the initial nectar source location, x min,i and x max,i are the lower and upper bounds of the optimization design variable of the i-th dimension, and rand is a random number between (0,1); S7.2: Substitute the objective function in the mechanical system reliability optimization design model into the fitness function as the target and calculate the fitness. The fitness function expression is: Among them: fit i is the fitness value, f i is the objective function, i represents the sample number, and n represents the number of samples; S7.3: Record the current optimal fitness value S7.4: Introduce adaptive step size to update the nectar source position, and then iterate the optimal fitness value. The expression for updating the nectar source position is: Where: v i is the update speed, x best,i is the current optimal variable value, represents the i-th random number of the i-th x, represents the jth random number of the ith x, Step is the adaptive step size, and the expression of the adaptive step size is: Among them: Cycle is the current number of iterations, max Cycle is the maximum number of iterations set by the algorithm; S7.5: Use the improved roulette method to compare the optimization results, and the expression is: Where: p i Represents the probability value of roulette, max(fit i ) is the maximum value of fitness, S7.6: Compare the number reached by the counter trail with the maximum number of optimization times N. If trail ≥ N, then stop updating the nectar source location, and output the optimization result of the optimal optimization design variable according to the current optimal fitness. If trail < N, then continue to update the nectar source location. The expression is: Where: v i is the update speed, L min,i is the lower limit of the i-th random number, U max,i is the upper limit of the i-th random number, x i is the optimization design variable corresponding to the current nectar source location.

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