Structural reliability optimization design method based on probability and probability box hybrid model

Abstract

The invention discloses a structural reliability optimization design method based on a probability and probability box hybrid model, and the method comprises the steps: building a solving model of a reliability index through employing a quadratic four-moment method based on a maximum entropy principle, so as to improve the calculation efficiency of reliability analysis; the classification capability of a K-nearest neighbor algorithm model is adopted to replace a reliability evaluation process in constraint conditions, and design variables are classified so as to reduce the calculation cost of the constraint conditions in the optimization design process; according to the method, a complex nested optimization design problem can be converted into a single-layer deterministic optimization design problem, the optimization calculation efficiency can be improved essentially, and the method has wide engineering application value in the field of engineering mechanical structure design.

Description

Technical Field

[0001] This invention relates to the field of engineering machinery structural design technology, specifically to a structural reliability optimization design method based on a hybrid probability and probability box model. Background Technology

[0002] Uncertainty is prevalent in the optimization design of engineering structures, and reliability analysis can fully account for the impact of uncertainties. Therefore, applying reliability optimization design to the design of engineering machinery structures has extremely important research significance.

[0003] The existing engineering machinery structural design methods still have the following shortcomings:

[0004] 1. Existing engineering machinery structural design methods typically treat material parameters, load parameters, and design parameters as deterministic parameters. However, in actual operation, many parameters, such as random loads, non-homogeneous materials, and geometric dimensions, exhibit random and cognitive uncertainties. Treating all of these as deterministic parameters will cause a deviation between the product's actual and expected working capabilities. As the product's functions and structures become increasingly complex, this deviation will be further amplified, potentially leading to unnecessary accidents in the engineering machinery.

[0005] 2. In the field of uncertainty analysis, both stochastic uncertainty and cognitive uncertainty have played important roles and yielded many research results. However, in actual engineering problems, stochastic uncertainty and cognitive uncertainty usually coexist, making it impossible to obtain the precise distribution of certain parameters. This limits the applicability of traditional probability models in the field of engineering machinery structure design. A model that can describe multiple uncertainty problems is needed so that the optimization results can be closer to the actual situation.

[0006] 3. Currently, most structural reliability optimization design methods for engineering machinery structures can only handle single-objective optimization. However, actual engineering structures may require simultaneous optimization of multiple objectives. Therefore, single-objective optimization methods cannot fully meet design needs. Furthermore, the reliability analysis process of constraints often involves complex double-layer nested optimization, thus requiring triple-layer nested optimization for multi-objective optimization design of reliability-based mechanical structures. Direct solution would inevitably lead to an unbearable computational burden.

[0007] In summary, improving the computational efficiency of decoupling multi-layered nested optimization problems and employing appropriate models to solve the uncertainties in the reliability optimization design process of engineering machinery structures are currently key to the reliability optimization design of engineering machinery structures. Furthermore, there are currently no patents in engineering machinery structure reliability optimization design methods that simultaneously employ hybrid models to describe the uncertainties in the optimization design process and decouple multi-layered nested optimization problems. Therefore, developing a hybrid model-based reliability optimization design method for engineering machinery structures can not only improve the working performance of engineering machinery structures but also has significant implications for the safety of engineering structures. Summary of the Invention

[0008] To overcome the above problems, this invention proposes a structural reliability optimization design method based on a hybrid probability and probability box model to solve the above-mentioned multiple problems.

[0009] The technical solution adopted by this invention to solve its technical problem is: a structural reliability optimization design method based on a hybrid probabilistic and probabilistic box model, characterized by comprising the following steps:

[0010] Step 1: Define the design variables, random variables, probability box variables, objective function, and constraint functions for the reliability optimization design problem. Establish a multi-objective reliability optimization design mathematical model based on a hybrid probability and probability box model. The specific expression is as follows:

[0011]

[0012] In the formula, f s (d,X,Y) is the objective function, G j (d,X,Y) are constraint functions. For the reliability index to be determined, The target reliability index is represented by d, and the k-dimensional design variables have a range of values. X represents an n-dimensional random variable, whose range of values ​​is: Y represents an m-dimensional probability box variable, whose value range is: μ X Let μ be the expected value of a random vector X. x Let Y be the nominal value of the probability box vector.

[0013] Step 2: Randomly sample the design variables in the entire deterministic design space using the Latin hypercube experimental design method to obtain a sample of design variables {d}. (1) d (2) ,···,d (N)};

[0014] Step 3: Decouple the reliability optimization design problem based on the probability and probability box hybrid model using the maximum entropy principle, and obtain all sample points {d}. (1) d (2) ,···,d (N) The corresponding minimum reliability index Decoupling the reliability optimization design problem includes the following steps:

[0015] Step 31: First, for the limit state function g(d) (i) The transformation is obtained by standardizing (X,Y), and the specific transformation is shown in formula (11):

[0016]

[0017] In the formula, Z′ represents a standard normal random variable. and Let represent the mean and standard deviation of the limit state function, respectively;

[0018] Step 32: Calculate the first four central moments of the standard normal random variable Z′ using the following formula, with the specific transformation shown in formula (12):

[0019]

[0020] In the formula, and Let g(d) represent the limit state function respectively. (i) The third and fourth central moments of (x,y);

[0021] Step 33: Calculate the first four central moments of the standard random variable Z′. As a constraint, the analytical expression ρ(Z′) of its probability density function can be obtained, and the specific transformation is shown in formula (13):

[0022]

[0023] In the formula, λ i (i=0,1,…,4) represents the Lagrange multipliers, which are a set of undetermined coefficients that can be obtained from the nonlinear equation (14):

[0024]

[0025] Step 34: Its failure probability can be calculated using formula (15):

[0026]

[0027] Step 35: Since the distribution parameter θ of the probability box variable Y is given in interval form, the following single-layer optimization model can be established:

[0028]

[0029] Step 4: Based on the set reliability index threshold For each sample d (i) The data (i = 1, 2, ..., n) are classified, and then labeled according to the categories to construct the training set of the K-nearest neighbor algorithm model. The specific classification is shown in formula (17):

[0030]

[0031] Step 5: Initial selection of K-nearest neighbor algorithm model K value (K={K1,K2,···,K i The K-nearest neighbor algorithm model was evaluated and the K value was selected using a 10-fold cross-validation method.

[0032] Step 6: Initialize the multi-objective genetic algorithm and set its relevant parameters: maximum number of iterations Ι max Population size p s Crossover probability p c , convergence tolerance ξ;

[0033] Step 7: Optimize the design variable d using a multi-objective genetic algorithm and calculate the objective function.

[0034] Step 8: Call the K-Nearest Neighbors algorithm model to calculate the relationship between the design variable d and each sample d. (i) The Euclidean distances for (i = 1, 2, ..., n) are shown in formula (18):

[0035] dist(d * d (i) )=||d * -d (i) ||2, i=1,2,…,N Formula (18)

[0036] In the formula, d * For the sample to be tested, d (i) (i = 1, 2, ..., n) are deterministic samples;

[0037] Step 9: Determine whether d satisfies the reliability constraint based on the categories of the K nearest samples;

[0038] Step 10: Determine the objective function value between the two generations. Or solution set d * Is the change less than the set convergence tolerance ξ? If it is less, proceed to the next step; otherwise, return to step 8.

[0039] Step 11: Output the optimal Pareto solution set d of the design parameters. * and objective function value

[0040] Preferably, the objective function and constraint function in step 1 are approximated by using a surrogate model, which is constructed using a kriging function.

[0041] Preferably, in step 3, a genetic algorithm is used as the optimization solver to obtain the current optimal solution β. (i)L .

[0042] The beneficial effects of this invention are:

[0043] 1. In response to point 1 in the background art, the present invention uses a hybrid probabilistic model and a probabilistic box model to describe some uncertain parameters in the design of engineering machinery structures, thereby effectively reducing the deviation between the actual working performance and the expected working performance of engineering machinery structures, and effectively ensuring the safety of engineering machinery structures.

[0044] 2. Regarding point 2 in the background technology, this invention uses a hybrid probability and probability box model to measure the uncertainty variables in engineering machinery structures. For uncertainty variables with sufficient data information, the uncertainty is measured by a probability model, while for uncertainty variables whose precise distribution cannot be constructed, the probability box model is used to describe them. This makes the optimization results closer to the actual situation and more in line with the design purpose of actual engineering machinery structures.

[0045] 3. Regarding point 3 in the background technology, this invention proposes a structural reliability optimization design method based on a hybrid probability and probability box model, which can be used to solve multi-objective optimization problems in engineering machinery. It employs a quadratic fourth-order moment method based on the maximum entropy principle and a K-nearest neighbor algorithm model to transform the original three-layer nested multi-objective reliability optimization design problem into a single-layer optimization problem, successfully decoupling the nested optimization problem. Simultaneously, it uses the Kriging function to construct the corresponding approximate reliability optimization design problem and employs a genetic algorithm for solving it, effectively improving computational efficiency.

[0046] Note: The above designs are not in any particular order, and each one makes the present invention different from the prior art and a significant advancement. Attached Figure Description

[0047] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0048] Figure 1 This is a flowchart of the structural reliability optimization design method based on the probability and probability box hybrid model of the present invention;

[0049] Figure 2 A schematic diagram of the simulation model of the upper structure of the bucket wheel stacker-reclaimer in a specific embodiment.

[0050] The reference numerals in the figure are as follows:

[0051] 1. Bucket wheel excavator front boom; 2. Bucket wheel excavator front tie rod; 3. Gantry structure; 4. Counterweight tie rod; 5. Counterweight boom. Detailed Implementation

[0052] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings;

[0053] like Figure 1 As shown, a structural reliability optimization design method based on a hybrid probabilistic and probabilistic box model includes the following steps:

[0054] Step 1: Define the design variables, random variables, probability box variables, objective function, and constraint functions for the reliability optimization design problem. Establish a multi-objective reliability optimization design mathematical model based on a hybrid probability and probability box model. The specific expression is as follows:

[0055]

[0056] In the formula, f s (d, X, Y) is the objective function, G j (d, X, Y) are constraint functions. For the reliability index to be determined, The target reliability index is represented by d, and the k-dimensional design variables have a range of values. X represents an n-dimensional random variable, whose range of values ​​is: Y represents an m-dimensional probability box variable, whose value range is: μ X Let μ be the expected value of a random vector X. X Let Y be the nominal value of the probability box vector.

[0057] Step 2: Randomly sample the design variables in the entire deterministic design space using the Latin hypercube experimental design method to obtain a sample of design variables {d}. (1) d (2) ,···,d (N)};

[0058] Step 3: Decouple the reliability optimization design problem based on the probability and probability box hybrid model using the maximum entropy principle, and obtain all sample points {d}. (1) d (2) ,···,d (N) The corresponding minimum reliability index Decoupling the reliability optimization design problem includes the following steps:

[0059] Step 31: First, for the limit state function g(d) (i)The transformation is obtained by standardizing (X,Y), and the specific transformation is shown in formula (20):

[0060]

[0061] In the formula, Z′ represents a standard normal random variable. and Let represent the mean and standard deviation of the limit state function, respectively;

[0062] Step 32: Calculate the first four central moments of the standard normal random variable Z′ using the following formula, with the specific transformation shown in formula (21):

[0063]

[0064] In the formula, and Let g(d) represent the limit state function respectively. (i) The third and fourth central moments of (x,y);

[0065] Step 33: Calculate the first four central moments of the standard random variable Z′. As a constraint, the analytical expression ρ(Z′) of its probability density function can be obtained, and the specific transformation is shown in formula (23):

[0066]

[0067] In the formula, λ i (i=0,1,…,4) represents the Lagrange multipliers, which are a set of undetermined coefficients that can be obtained from the nonlinear equation (25):

[0068]

[0069] Step 34: Its failure probability can be calculated using formula (6):

[0070]

[0071] Step 35: Since the distribution parameter θ of the probability box variable Y is given in interval form, the following single-layer optimization model can be established:

[0072]

[0073] Step 4: Based on the set reliability index threshold For each sample d (i) The data (i = 1, 2, ..., n) are classified, and then labeled according to the categories to construct the training set of the K-nearest neighbor algorithm model. The specific classification is shown in formula (29):

[0074]

[0075] Step 5: Initial selection of K-nearest neighbor algorithm model K value (K={K1,K2,···,K i The K-nearest neighbor algorithm model was evaluated and the K value was selected using a 10-fold cross-validation method.

[0076] Step 6: Initialize the multi-objective genetic algorithm and set its relevant parameters: maximum number of iterations Ι max Population size p s Crossover probability p c , convergence tolerance ξ;

[0077] Step 7: Optimize the design variable d using a multi-objective genetic algorithm and calculate the objective function.

[0078] Step 8: Call the K-Nearest Neighbors algorithm model to calculate the relationship between the design variable d and each sample d. (i) The Euclidean distances for (i = 1, 2, ..., n) are shown in formula (30):

[0079] dist(d * d (i) )=||d * -d (i) ||2, i=1,2,…,N Formula (30)

[0080] In the formula, d * For the sample to be tested, d (i) (i = 1, 2, ..., n) are deterministic samples;

[0081] Step 9: Determine whether d satisfies the reliability constraint based on the categories of the K nearest samples;

[0082] Step 10: Determine the objective function value between the two generations. Or solution set d * Is the change less than the set convergence tolerance ξ? If it is less, proceed to the next step; otherwise, return to step 8.

[0083] Step 11: Output the optimal Pareto solution set d of the design parameters. * and objective function value

[0084] To further illustrate the present invention in detail, a specific embodiment will be used to describe the solution of the present invention below. This embodiment takes the optimized design of the upper structure of a bucket wheel stacker-reclaimer as an example, and is implemented under the premise of the technical solution of the present invention. Detailed implementation methods and specific operating procedures are given, but the scope of protection of the present invention is not limited to the following embodiment.

[0085] like Figure 2The image shown is a simulation model of the upper structure of a bucket wheel stacker-reclaimer according to a specific embodiment. Figure 1 The process shown illustrates a multi-objective reliability optimization design for the superstructure of a bucket wheel stacker-reclaimer. The specific steps are as follows:

[0086] Step 1: Establish a numerical simulation model of the upper structure of the bucket wheel stacker-reclaimer. To improve the working performance of the upper structure, the total deformation L and overall mass M of the upper structure are selected as the dual objectives for optimization. The maximum Von Mises stress within the upper structure is set to not exceed the stress threshold of 300 MPa as a reliability constraint, aiming to minimize the mass of the upper structure while meeting reliability requirements, thus achieving a lightweight design. Considering that the working performance of the upper structure is mainly related to the steel plate thickness d1 of the portal frame and the steel plate thickness d2 of the counterweight boom, these two parameters are selected as deterministic design variables. Simultaneously, considering the randomness of the working environment, the front tie rod radius X1 and the counterweight tie rod radius X2 are used as probability variables, and the digging force Y1 of the bucket wheel is used as a probability box variable. The following multi-objective reliability optimization design mathematical model for the upper structure of the bucket wheel stacker-reclaimer, based on a hybrid probability and probability box model, can be established:

[0087] L(d1,d2,X1,X2,Y1) stβ L [g(d1,d2,X1,X2,Y1)=300-σ(d1,d2,X1,X2,Y1)≥0]≥β t d = (d1, d2) T 6mm≤d1≤18mm 5mm≤d2≤15mm, X1~N(μ) X1 ,σ X1 )

[0088] In the formula, β L β represents the reliability index to be determined. t This represents the set target reliability index. The front tie rod radius X1 follows a normal distribution, and its expected value is... Standard deviation The radius X2 of the counterweight rod follows a normal distribution, and its expected value is... Standard deviation The digging force Y1 follows an extreme value type 1 distribution, and its nominal value of the location parameter is... The nominal value of its scale parameter

[0089] By using the Kriging function to construct approximate models of the objective function and constraint functions, an approximate multi-objective reliability optimization design mathematical model is established as shown in formula (2):

[0090] d = (d1, d2) T 6mm≤d1≤18mm

[0091] In the formula, and For the approximate objective function, It is an approximate limit state function;

[0092] Step 2: Randomly sample the design variables in the entire deterministic design space using the Latin hypercube experimental design method to obtain a sample of design variables. and

[0093] Step 3: Decouple the reliability optimization design problem based on probability and probability box hybrid models by using the maximum entropy principle, and obtain all sample points. and The corresponding minimum reliability index {β} (1)L ,β (2)L , ···, β (N)L The decoupling of the reliability optimization design problem includes the following steps:

[0094] Step 31: First, consider the limit state function. The standardization is obtained, and the specific transformation is shown in formula (32):

[0095]

[0096] In the formula, Z′ represents a standard normal random variable. and Let represent the mean and standard deviation of the limit state function, respectively;

[0097] Step 32: Calculate the first four central moments of the standard normal random variable Z′ using the following formula, with the specific transformation shown in formula (32):

[0098]

[0099] In the formula, and Represent the limit state function respectively The third and fourth order central moments;

[0100] Step 33: Calculate the first four central moments of the standard random variable Z′. As a constraint, the analytical expression ρ(Z′) of its probability density function can be obtained, and the specific transformation is shown in formula (33):

[0101]

[0102] In the formula, λ i (i=0,1,…,4) represents the Lagrange multipliers, which are a set of undetermined coefficients that can be obtained from the nonlinear equation (34):

[0103]

[0104] Step 34: Its failure probability can be calculated using formula (35):

[0105]

[0106] Formula (35)

[0107] Step 35: Due to the distribution parameters of the probability box variable Y1 Given in interval form, the following single-layer optimization model can be established:

[0108]

[0109] Step 4: Based on the set reliability index threshold β t =6, for each sample d (i) The data (i = 1, 2, ..., N) are classified, and then labeled according to the categories to construct the training set of the K-nearest neighbor algorithm model. The specific classification is shown in formula (37):

[0110]

[0111] Step 5: Initial selection of K-nearest neighbor algorithm model K value (K={K1,K2,···,K i The K-nearest neighbor algorithm model was evaluated and the K value was selected using a 10-fold cross-validation method.

[0112] Step 6: Initialize the multi-objective genetic algorithm and set its relevant parameters: maximum number of iterations Ι max =600, population size p s =50, crossover probability p c =0.8, convergence tolerance ξ=0.001;

[0113] Step 7: Optimize the design variable d using a multi-objective genetic algorithm and calculate the objective function.

[0114] Step 8: Call the K-Nearest Neighbors algorithm model to calculate the relationship between the design variable d and each sample d. (i) The Euclidean distance for (i = 1, 2, ..., n) is shown in formula (38):

[0115] dist(d * d (i) )=||d * -d (i) ||2, i=1,2,…,N Formula (38)

[0116] In the formula, d * For the sample to be tested, d (i) (i = 1, 2, ..., n) are deterministic samples;

[0117] Step 9: Determine whether d satisfies the reliability constraint based on the categories of the K nearest samples;

[0118] Step 10: Determine the objective function value between the two generations. Or solution set d * Is the change less than the set convergence tolerance ξ? If it is less, proceed to the next step; otherwise, return to step 8.

[0119] Step 11: Output the optimal Pareto solution set d of the design parameters. * and objective function value

[0120] This embodiment converged after 229 genetic iterations, ultimately obtaining the Pareto optimal solution set shown in Table 1. As shown in Table 1, through multi-objective reliability optimization design of the steel plate thickness of the portal frame structure and the steel plate thickness of the counterweight boom of the bucket wheel stacker-reclaimer's upper structure, a total of 18 optimal solutions for design parameters were obtained. In other words, when it is necessary to achieve a lightweight design of the upper structure of the bucket wheel stacker-reclaimer while meeting the requirements of safety and reliability, the following design schemes can be selected.

[0121] Table 1 Pareto solution set for the optimization problem of the superstructure of the bucket wheel stacker-reclaimer.

[0122]

[0123] The above detailed description is a specific description of feasible embodiments of the present invention. These embodiments are not intended to limit the patent scope of the present invention. All equivalent implementations or modifications that do not depart from the present invention should be included in the patent scope of this case.

Claims

1. A structural reliability optimization design method based on a hybrid probabilistic and probabilistic box model, characterized in that, Includes the following steps: Step 1: Define the design variables, random variables, probability box variables, objective function, and constraint functions for the reliability optimization design problem. Establish a multi-objective reliability optimization design mathematical model based on a hybrid probability and probability box model. The specific expression is as follows: s.t. β j L [g j (d,X,Y)≥0]≥β t j ,j=1,2,···,q In the formula, f s (d, X, Y) is the objective function, G j (d, X, Y) are constraint functions. For the reliability index to be determined, The target reliability index is represented by d, and the k-dimensional design variables have a range of values. X represents an n-dimensional random variable, whose range of values ​​is: Y represents an m-dimensional probability box variable, whose value range is: μ X Let μ be the expected value of a random vector X. X Let Y be the nominal value of the probability box vector. Step 2: Randomly sample the design variables in the entire deterministic design space using the Latin hypercube experimental design method to obtain a sample of design variables {d}. (1) d (2) ,···,d (N) }; Step 3: Decouple the reliability optimization design problem based on the probability and probability box hybrid model using the maximum entropy principle, and obtain all sample points {d}. (1) d (2) ,···,d (N) The corresponding minimum reliability index Decoupling the reliability optimization design problem includes the following steps: Step 31: First, for the limit state function g(d) (i) The transformation is obtained by standardizing (X,Y), and the specific transformation is shown in formula (2): In the formula, Z′ represents a standard normal random variable. and Let represent the mean and standard deviation of the limit state function, respectively; Step 32: Calculate the first four central moments of the standard normal random variable Z′ using the following formula, with the specific transformation shown in formula (3): In the formula, and Let g(d) represent the limit state function respectively. (i) The third and fourth central moments of (x,y); Step 33: Calculate the first four central moments of the standard random variable Z′. As a constraint, the analytical expression ρ(Z′) of its probability density function can be obtained, and the specific transformation is shown in formula (4): In the formula, λ i (i=0,1,…,4) represents the Lagrange multipliers, which are a set of undetermined coefficients that can be obtained from the nonlinear equation (5): Step 34: Its failure probability can be calculated using formula (6): Step 35: Since the distribution parameter θ of the probability box variable Y is given in interval form, the following single-layer optimization model can be established: Step 4: Based on the set reliability index threshold For each sample d (i) The data (i = 1, 2, ..., n) are classified, and then labeled according to the categories to construct the training set of the K-nearest neighbor algorithm model. The specific classification is shown in formula (8): Step 5: Initial selection of K-nearest neighbor algorithm model K value (K={K1,K2,···,K i The K-nearest neighbor algorithm model was evaluated and the K value was selected using a 10-fold cross-validation method. Step 6: Initialize the multi-objective genetic algorithm and set its relevant parameters: maximum number of iterations Ι max Population size p s Crossover probability p c , convergence tolerance ξ; Step 7: Optimize the design variable d using a multi-objective genetic algorithm and calculate the objective function. Step 8: Call the K-Nearest Neighbors algorithm model to calculate the relationship between the design variable d and each sample d. (i) The Euclidean distance for (i = 1, 2, ..., n) is shown in formula (9): In the formula, d * For the sample to be tested, d (i) (i = 1, 2, ..., n) are deterministic samples; Step 9: Determine whether d satisfies the reliability constraint based on the categories of the K nearest samples; Step 10: Determine the objective function value between the two generations. Or solution set d * Is the change less than the set convergence tolerance ξ? If it is less, proceed to the next step; otherwise, return to step 8. Step 11: Output the optimal Pareto solution set d of the design parameters. * and objective function value 2. The structural reliability optimization design method based on a hybrid probability and probability box model according to claim 1, characterized in that: The objective function and constraint function in step 1 are approximated by using a surrogate model, which is constructed using a kriging function.

3. The structural reliability optimization design method based on a hybrid probability and probability box model according to claim 1, characterized in that: In step 35, a genetic algorithm is used as the optimization solver to obtain the current optimal β. (i)L .

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