First-order reliability method for hybrid support vector regression and global optimal particle swarm optimization of ZnO nanocomposite beam

Abstract

The invention discloses a first-order reliability method for hybrid support vector regression and global optimal particle swarm optimization of a ZnO nano composite material beam, which is applied to the technical field of reliability analysis of a nano composite material beam structure and adopts a hybrid machine learning strategy combining the first-order reliability method and the support vector regression. Meanwhile, a method based on global optimal particle swarm optimization is integrated, and a search method which is high in robustness and is used for determining the precision and the efficiency based on the performance most probable point (MPP) is provided; according to the method, the calculation precision and calculation efficiency of reliability can be effectively improved.

Description

Technical Field

[0001] The invention belongs to the technical field of structural reliability analysis, and particularly relates to a first-order reliability analysis technology. Background Art

[0002] Reliability refers to the ability of a product to perform its intended functions under specified conditions. Higher reliability means lower probability of failure. For a complex mechanical system, it is particularly important to maintain high reliability, because once a failure occurs, it may have catastrophic consequences and unbearable costs.

[0003] The reliability of ZnO nanocomposite beam structures involves multi-source uncertainties caused by materials, manufacturing processes, dimensions, strength, and loads. In statistical or probabilistic analysis, multi-source uncertainties are evaluated using probabilistic models, where performance functions are used in structural reliability analysis to approximate reliability through failure probabilities.

[0004] Machine learning methods have played a huge role since they were introduced. Support vector regression (SVR) can predict performance functions with efficient training points using a small database. However, the analytical iteration of the first order reliability method (FORM) may not converge for all cases on the approximate function using SVR. Therefore, the balance between the MPP (Most probable point, performance-based most likely point) search and the accurate prediction of SVR is a key factor in hybrid reliability analysis. Summary of the invention

[0005] In order to solve the above technical problems, the present invention proposes a first-order reliability method of hybrid support vector regression and global optimal particle swarm optimization for ZnO nanocomposite beams, which can better balance the reliability analysis of hybrid structures, reduce the amount of calculation in the analysis, and improve the calculation accuracy and efficiency of the reliability of beams.

[0006] The technical solution adopted by the present invention is: a first-order reliability method of hybrid support vector regression and global optimal particle swarm optimization for ZnO nanocomposite beams, comprising:

[0007] S1. Define the reliability problem; according to the physical properties of the ZnO nanocomposite beam structure, load conditions and other factors, combined with specific engineering application requirements and historical experience, set the upper and lower limits of the basic random variables in the reliability problem to be analyzed. The basic random variables here include: beam height, beam length, basic spring constant, basic shear constant, volume fraction of ZnO, applied voltage and other variables. According to the characteristics of the search space and search samples, balance the speed of global search and local search in specific problems, and set the basic search speed of the particle swarm. For example, when the nonlinearity of the limit state function is high, it is necessary to set a lower initial speed to prevent falling into the local optimum.

[0008] S2, conduct simulation process; firstly, give the sample parameters NS of the basic random variables used in the structural reliability analysis of ZnO nanocomposite beams, including the height of the beam, the length of the beam, the basic spring constant, the basic shear constant, the volume fraction of ZnO and the applied voltage, etc. These sample parameters are given according to the upper and lower limits of the basic random variables. Then, according to the given random variable sample parameters, Latin hypercube sampling (LHS) is used to generate basic random variable sample points, and finally the limit state function (LSF) is calculated.

[0009] S3. Modeling through SVR: First, given the hyperparameters C, ε, σ in SVR, by considering K(X,X i ) obtains the N-dimensional input with a regular form, obtains the optimal weight by optimizing the loss function of the SVR model as shown in formula (1), and trains the SVR model with the data set composed of the basic random variable sample points generated in S2.

[0010]

[0011] where y i It represents the response value of the structural system performance, which means the response data obtained through observation and used to compare with the predicted function f(X i ,w) to compare and obtain the residual of the calculation model for optimizing the model calculation.

[0012] Finally, the SVR model can be expressed by formula (2):

[0013]

[0014] S4. Debugging of hyperparameters in the SVR model; First, set the optimization factors, which specifically include: the partial size (PS), the number of variables (NV), and the total number of iterations (NI), and define the initial number of iterations as k = 1. PS represents how many particles (i.e., the search points participating in the optimization process in each iteration, which is the basic unit in the particle swarm optimization algorithm) participate in the optimization process in each iteration, and its value depends on the problem complexity and the search space dimension; NV represents the dimension of the problem, that is, the number of variables to be optimized, depending on the optimization objective in the specific analysis; NI represents the total number of times the optimization algorithm runs, depending on the convergence speed of the algorithm and the computing resources. In the SVR model, assume that PS is 20, that is, 20 particles search for the optimal solution in space in each iteration. NV is 3, that is, the hyperparameters C, ε, σ to be optimized in the SVR, and NI is the total number of iterations set according to the actual iteration needs.

[0015] Then determine the corresponding hyperparameter ranges for the SVR under the PS dimension, which are 5 < C < 50000, 0.001 < ε < 0.2, 0.5 < σ < 5. According to the model influence corresponding to the hyperparameters, in order to prevent overfitting, smaller hyperparameter values can be taken, such as C = 20, ε = 0.05, σ = 1. Based on this, iterate for the case of k < NI, and use g best Give the best hyperparameters C, ε, σ of the corresponding SVR, and construct the SVR model using the best hyperparameters. The SVR model constructed using the best hyperparameters can be expressed as formula (3).

[0016]

[0017] S5. Search for the MPP point through high-performance SVR-GPSO; Based on the optimization factors, the number of variables, the total number of iterations, and the best hyperparameters set in S4, enter the MPP search process. When k < NI, first transform the random variable from the X space to the U space U i = Φ -1 [F x (x i )], then obtain the deterministic enhanced probability model, update and and obtain the new MPP, stop when iterating to convergence or the maximum number of times, and output the reliability index β.

[0018] Advantages of the present invention: Compared with the prior art, the present invention provides a first-order reliability method for a ZnO nanomaterial composite beam using hybrid support vector regression and global best particle swarm optimization, and has the following advantages:

[0019] (1) Compared with the iterative method HL-RF and the directional stability transformation method DSTM, the meta-heuristic methods such as PSO and GPSO provided by the present invention and the FLS method that uses sufficient descent conditions to adjust the step size have stable results.

[0020] (2) Compared with the traditional probability model of FORM, the enhanced probability model based on fault domain MPP search provided by the present invention provides an accurate reliability index for the reliability problem of ZnO nanocomposite beams.

[0021] (3) The present invention adjusts the global optimal particles through a local search framework, thereby enhancing the globality of the particle swarm method.

[0022] (4) The proposed SVR based on intelligent optimization method combined with MPP search based on enhanced probability model can improve the efficiency of reliability analysis of implicit engineering problems of ZnO nanocomposite beams.

[0023] (5) By using two loss functions to adjust the hyperparameters of SVR and training SVR with particle swarm method or hybrid training of particle swarm method, the accuracy of SVR in MPP search is improved.

[0024] (6) In the reliability analysis of ZnO nanocomposite beams, the results of SVR-GPSO provided by the present invention are more accurate than SVR and more effective than analytical methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 Analyze the process for SVR-GPSO;

[0026] Figure 2 It is the GPSO method flow;

[0027] Figure 3 Comparison of the results of different reliability analysis methods in the case. DETAILED DESCRIPTION

[0028] To facilitate those skilled in the art to understand the technical content of the present invention, the present invention is further explained below with reference to the accompanying drawings.

[0029] like Figure 1 As shown, a first-order reliability method for a ZnO nanocomposite beam using a hybrid support vector regression and global optimal particle swarm optimization of the present invention comprises the following steps:

[0030] S1. Define the reliability problem; according to the physical properties of the ZnO nanocomposite beam structure, load conditions and other factors, combined with specific engineering application requirements and historical experience, set the upper and lower limits of the basic random variables in the reliability problem to be analyzed. The basic random variables here include: beam height, beam length, basic spring constant, basic shear constant, volume fraction of ZnO, applied voltage and other variables. According to the characteristics of the search space and search samples, balance the speed of global search and local search in specific problems, and set the basic search speed of the particle swarm. For example, when the nonlinearity of the limit state function is high, it is necessary to set a lower initial speed to prevent falling into the local optimum.

[0031] S2, conduct simulation process; firstly, give the sample parameters NS of the basic random variables used in the structural reliability analysis of ZnO nanocomposite beams, including the height of the beam, the length of the beam, the basic spring constant, the basic shear constant, the volume fraction of ZnO and the applied voltage, etc. These sample parameters are given according to the upper and lower limits of the basic random variables. Then, according to the given random variable sample parameters, Latin hypercube sampling (LHS) is used to generate basic random variable sample points, and finally the limit state function (LSF) is calculated.

[0032] S3. Perform SVR modeling. The main problem of SVR is to calculate a suitable regression function, which is a regression function with a small residual between the observed response (y) of the structural system performance in the ZnO nanocomposite beam to be analyzed and the predicted f(X,w), where (y) is the observed system response y i For the input data set [(X1,y1),(X2,y2),...,(X N ,y N )], SVR is used to find the relationship between the response (y) as the performance of the structural system and the input random variables (X) as the multi-source uncertainty. By using the input data set as the response (y) and (X), SVR-based machine learning transforms the n input random variables from X∈R through the function f(X,w) n The space is transformed into g∈R space, as shown below:

[0033] g=[f(X,w),w∈Λ|f:R n →R] (4)

[0034] Among them, Λ is the domain of the unknown coefficient as the weight w of the SVR model that needs to be trained; f represents the regression function LSF (Limit state function) used to map n input variables to 1 output response; by converting the input random variable X into the output response g, SVR can effectively approximate the limit state function of the structure.

[0035] In SVR, an ε-insensitive loss function is given for the training model, which obtains a smaller error (e) by adjusting the weights during the regression process as follows:

[0036]

[0037] Where ε represents the insensitivity value applied in the loss function. In the case of ε>0, by calculating the prediction function f(X i ,w) and the observed response y i The error between them can be ignored when it is less than ε.

[0038] At this time, the SVR model can be expressed by the following formula:

[0039]

[0040] Where b is the deviation, w i Represents X i The corresponding weight, K(X,X i ) is the kernel function. The kernel function is used as the nonlinear mapping relationship, which can be expressed as follows:

[0041] K(X,X i )=exp(-0.5||X,X i || 2 / σ 2 ) (7)

[0042] Among them, σ is the most commonly used core parameter, and the kernel function can be calculated using the core parameters to cover an infinite domain.

[0043] Combining formula (5) with formula (6), consider K(X,X i ) is obtained in the form of a rule, and formula (5) is rewritten as formula (8). The SVR model is trained with a data set consisting of basic random variable sample points generated by LHS, and the optimal weight is obtained through the loss function model shown in formula (8):

[0044]

[0045] Among them, C ≥ 0 is the regularization coefficient.

[0046] The prediction results of SVR depend on the hyperparameters in the optimization model (8), namely, C, ε, and σ, whose value ranges are 5 <C<50000,0.001<ε<0.2,0.5<σ<5。

[0047] S4. In the reliability analysis of the invention, the absolute loss function is used to adjust the hyperparameters of SVR, which can be expressed by the following formula:

[0048]

[0049] Where N represents the number of basic random variable sample points used in the training phase and in constructing the kernel function.

[0050] The process of hyperparameter tuning is:

[0051] A1. First, it is necessary to define the basic reliability problem according to the basic structural characteristics of the ZnO nanocomposite beam to be analyzed, the specific application scenario, and the main random variables affecting its structure. The basic FORM model is as follows:

[0052]

[0053] Among them, U is the standard normal random variable projected into its space by X, that is, the vector of basic random variables in the structural reliability of ZnO nanocomposite beams, including the height of the beam, the length of the beam, the basic spring constant, the basic shear constant, the volume fraction of ZnO, the applied voltage and other variables. The corresponding relationship is U i =Φ -1 [F x (x i )], that is, the process of transforming the original random variable X into a standard normal random variable U, Φ -1 is the inverse function of the standard normal distribution function, x i are the components of vector X.

[0054] The goal of this FORM model is to find the minimum length ||U|| from the origin of the form to the limit state function g(U) in the ordinary standard normal space. This point is the MPP point searched in the method of the present invention. Its basic iteration is as follows:

[0055]

[0056] Among them U k is the result of the kth iteration, g(U k ) represents the limit state function corresponding to the result of the kth iteration, To find the gradient factor, α k is the normalized sensitivity vector, β k is the reliability index under the corresponding number of iterations.

[0057] In the basic FORM model of formula (10), a gradient solution is used to search for MPP on the limit state surface. According to the non-gradient solution, the search area can be expanded to consider a wider failure area, and the constraint condition in formula (10) is transformed from g(U)=0 to g(U)≤0. At this time, using the population search method provided by the present invention, the MPP can be searched by a penalty method, as shown in formula (12).

[0058] minf=||U||+λmax{0,g(U)} (12)

[0059] Where λ is the penalty factor, which can be expressed as g(μ) represents the value of the limit state function at the average value μ of the random variable. This value will directly affect the size of the penalty factor and is used to balance the relationship between the constraints and the objective function during the optimization process.

[0060] According to formula (10) and formula (12), the search basis of the present invention is to search for MPP based on the population search method, and its optimization model can be expressed by formula (13):

[0061] minf=||U||+λ|g(U)| (13)

[0062] A2. In order to improve the accuracy and efficiency of the FORM model, in searching for the MPP point, the present invention provides the following Figure 2 The global particle swarm optimization GPSO (Global-best particle swarm optimization) method shown in FIG. updates the new position of the particle as follows:

[0063]

[0064] where p best With g best are two particles that are the best population for all positions and the best population for the current position respectively; r, r1, and r2 are all random numbers, r is used to determine the overall movement direction of the particle, and r1 and r2 are used to adjust the particle's response to the historical best position; ω k is the inertia weight, which is used to control the movement of particles in the search space; V k+1 is the new search speed; X k+1 is the position of the new particle corresponding to the number of iterations of GPSO, that is, the variable result obtained after the k+1th iteration; γ k is the step length; δ k is the local search distance; P k Indicates the rate at which the global best particle is considered for adjustment.

[0065] in

[0066] in Where c1=c2=2;ω min =0.4,ω max =0.9; r, r1 and r2 are three random numbers between 0 and 1.

[0067] In GPSO, the search speed interval is defined as v max =0.5σ and v min =-0.5σ; the initial velocity (V0) is randomly generated from this interval. The GPSO pseudo code provided by the invention is shown in Table 1 below.

[0068] Table 1GPSO pseudo code

[0069]

[0070]

[0071] Under the above method conditions and The update iteration is continued until convergence or the maximum upper limit k = 300 times, and the final output is U best With X best , and the reliability index β=||U best ||.

[0072] This case 1 is an engineering example, a piezoelectric ZnO nanoparticle reinforced beam, its implicit performance function is expressed as:

[0073]

[0074] Among them, P cr is the critical buckling load, and P is the axial compression load, which is 55GPa.

[0075] Furthermore, the performance function of this problem is defined using the buckling failure theory, and the buckling force is calculated using the shear deformation theory. cr The implicit performance function of is evaluated using the six random variables listed in Table 2.

[0076] Table 2 Random variables of nanocomposite beams

[0077]

[0078] Furthermore, in this case, three methods, namely analytical method, simulation method and hybrid SVR, were compared. The indicators included the number of function calls (CF), the total number of iterations (NI), the robustness of the convergence reliability index β, and the accuracy (Relative Error, RE) obtained by the relative error between FORM and the Monte Carlo method.

[0079] Furthermore, Table 3 gives the comparison results of different reliability methods:

[0080] Table 3 Comparison results of different reliability methods

[0081]

[0082] In Table 1, HL-RF is the Hasofer-Lind Rackwitz-Flessler method; DSTM is the Directional Stability Transformation Method; TPM is the Traditional Probabilistic Model; and EPM is the Enhancing Probabilistic Model.

[0083] Furthermore, it can be concluded that the convergence reliability index β obtained by the SVR-GPSO of the present invention is very close to the result obtained by MCS (Monte Carlo Simulation) when the CF is the smallest, and the relative error is the smallest.

[0084] Further, Figure 3 The comparison results of the reliability indicators in the examples are presented, showing that the SVR-GPSO of the present invention has the most stable performance and the highest accuracy.

[0085] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, the present invention may have various changes and variations. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of the claims of the present invention.

Claims

1. A first-order reliability method for ZnO nanocomposite beams using hybrid support vector regression and global optimal particle swarm optimization, characterized in that include: S1. Determine the basic random variables of the ZnO nanocomposite beam structure and the corresponding upper and lower limits of the basic random variables; S2. Determine the sample parameter NS of the basic random variable according to the corresponding upper and lower limits of the basic random variable; then generate the sample points of the basic random variable through Latin hypercube sampling based on the sample parameter NS of the basic random variable, and finally calculate the limit state function corresponding to each sample point of the basic random variable; a basic random variable sample point and its corresponding limit state function are used as a training sample, thereby constructing a training sample set; S3, performing SVR modeling, training the SVR model based on the training sample set constructed in step S2; and also including obtaining the optimal hyperparameters of the SVR model based on a global particle swarm optimization method; S5. When the SVR model converges or reaches the maximum number of iterations, the reliability index is output.

2. A first-order reliability method for ZnO nanocomposite beams based on hybrid support vector regression and global optimal particle swarm optimization according to claim 1, characterized in that: The basic random variables include: beam height, beam length, basic spring constant, basic shear constant, volume fraction of ZnO, and applied voltage.

3. A first-order reliability method for ZnO nanocomposite beams based on hybrid support vector regression and global optimal particle swarm optimization according to claim 2, characterized in that: The SVR model is expressed as: Where b is the deviation, K(X,X i ) is the kernel function, X represents the set of basic random variables, X i represents the i-th basic random variable sample point generated by Latin hypercube sampling, w is the weight set of the SVR model, and N represents the number of active points used in the training phase and building the kernel function.

4. A first-order reliability method for ZnO nanocomposite beams using hybrid support vector regression and global optimal particle swarm optimization according to claim 3, characterized in that: In step S3, the SVR model is trained based on the training sample set constructed in step S2. The specific implementation process is as follows: A1. First, define the basic reliability issues corresponding to ZnO nanocomposite beams. The basic FORM model is as follows: Minimize||U|| subjected to g(U)=0 Among them, U is the standard normal random variable projected from X to its space, and its corresponding relationship is U i =Φ -1 [F x (x i )],Φ -1 is the inverse function of the standard normal distribution function, x i are the components of vector X; A2. Find the minimum length ||U|| from the origin to the limit state function g(U) in the ordinary standard normal space by iteration: Among them U k is the result of the kth iteration, g(U k ) represents the limit state function corresponding to the result of the kth iteration, To find the gradient factor, α k is the normalized sensitivity vector, β k is the reliability index under the corresponding number of iterations; A3. Change the constraint condition in the basic FORM model of step A1 from g(U)=0 to g(U)≤0; search the MPP point by penalty method based on population search method, as shown in the following formula: minf=||U||+λmax{0,g(U)} Among them, λ is the penalty factor, 5. A first-order reliability method for ZnO nanocomposite beams using hybrid support vector regression and global optimal particle swarm optimization according to claim 4, characterized in that: The optimal weights for the SVR model are obtained by combining the following loss function: Among them, y i Represents X i The corresponding limit state function calculated in step S2, C, ε are the hyper parameters of the SVR model.

6. A first-order reliability method for ZnO nanocomposite beams using hybrid support vector regression and global optimal particle swarm optimization according to claim 5, characterized in that: K(X,X i ) is: K(X,X i )=exp(-0.5||X,X i || 2 / σ 2 ) Among them, σ is the hyperparameter of the SVR model.

7. A first-order reliability method for ZnO nanocomposite beams using hybrid support vector regression and global optimal particle swarm optimization according to claim 6, characterized in that: In the process of searching for MPP points, the global particle swarm optimization method is used to update the particle positions: V k+1 =ω k V k +c1r1[p best -X k ]+c2r2[g best -X k ] Among them, p best With g best The two particles are the best population for all positions and the best population for the current position respectively; r, r1, and r2 are all random numbers. r is used to determine the overall movement direction of the particle, and r1 and r2 are used to adjust the particle's response to the historical best position; ω k is the inertia weight, which is used to control the movement of particles in the search space; V k+1 is the search speed of the k+1th iteration; V k is the search speed of the kth iteration; X k+1 is the position of the particle at the k+1th iteration; X k is the position of the particle at the kth iteration; γ k is the step length; δ k is the local search distance; P k Indicates the rate at which the global best particle is considered for adjustment.

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