Disc Spring Weight Optimization Method Based on Harris Hawk Algorithm with Random Unit Permutation

Abstract

The present invention discloses a method for optimizing the weight of a conical spring based on the Harris hawk algorithm with random unit permutation. First, the objective function for optimizing the weight of the conical spring and the key parameters to be solved are determined. Then, the structure of the original Harris hawk algorithm is optimized. The specific optimization measures are as follows: First, cancel the steps of centrally calculating the objective function value in the original Harris hawk algorithm except for the step of initializing the population, but record the results while the algorithm obtains a better solution. Second, introduce a random unit permutation mechanism at the position before the end of each iteration optimization in the original Harris hawk algorithm. Third, delete the search steps in the original Harris hawk algorithm where the absolute value of the energy factor E is greater than or equal to 1. The optimized Harris hawk algorithm is called the Harris hawk algorithm based on random unit permutation, and the key parameters are solved using this Harris hawk algorithm based on random unit permutation to obtain the key parameters for optimizing the weight of the conical spring. The advantages are fast convergence speed and high solution accuracy.

Description

Technical Field

[0001] The present invention relates to a method for optimizing the weight of a conical spring, and more particularly to a method for optimizing the weight of a conical spring based on a Harris hawks algorithm with random unit permutation. Background Art

[0002] The conical spring is abbreviated as a conical spring. It is named because it resembles a dish and has good buffering and shock absorption capabilities. It is widely used in scenarios that require vibration isolation, such as the elastic suspension of vehicles. The load-deformation characteristic of the conical spring shows a non-linear relationship, and its characteristic curve changes accordingly with the ratio of the inner truncated cone height to the spring steel plate thickness. In practical applications, it is necessary to optimize the weight of the conical spring. Due to its complex mathematical characteristics and constraint conditions, this task is regarded as an optimization problem with complex constraints.

[0003] Using a mathematical method to solve this problem is a relatively traditional method. However, this method is too dependent on the mathematical characteristics of the optimization model. When the model complexity increases, problems such as reduced solution accuracy, slow solution speed, and even unsolvability may occur. The Harris hawks algorithm is a novel meta-heuristic algorithm proposed in recent years, which can effectively alleviate the drawbacks of the mathematical method. The Harris hawks algorithm does not care about the specific mathematical characteristics of the model to be optimized and has the advantages of few parameters and strong generality, so it is widely used in the optimization field. However, the Harris hawks algorithm was not proposed to solve the problem of optimizing the weight of the conical spring. Therefore, when it is applied to the optimization of the conical spring weight, there are some defects in the solution process. First, the Harris hawks algorithm will repeatedly call the objective function during operation to adjust the algorithm search direction, resulting in the objective function at the same position being called multiple times, wasting computing resources. Second, the randomness of the global search stage of the Harris hawks algorithm is too high, resulting in too slow convergence speed of the algorithm. Finally, the lack of convergence of the local search stage of the Harris hawks algorithm results in too low solution accuracy of the algorithm. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for optimizing the weight of a conical spring based on a Harris hawks algorithm with random unit permutation, which has a fast convergence speed and high solution accuracy. The method for optimizing the weight of the conical spring optimizes the structure of the Harris hawks algorithm by introducing a random unit permutation mechanism into the original Harris hawks algorithm to obtain a Harris hawks algorithm with random unit permutation, extracts the key parameters for optimizing the weight of the conical spring using the Harris hawks algorithm with random unit permutation, further improves the solution speed and optimization accuracy, and finally obtains the optimal parameters on the premise of meeting the constraint conditions, so that the weight of the optimized conical spring structure is optimal.

[0005] The technical solution adopted by the present invention to solve the above technical problem is: A method for optimizing the weight of a conical spring based on a Harris hawks algorithm with random unit permutation, comprising the following steps:

[0006] Step S1: Determine the objective function for optimizing the weight of the disc spring and the key parameters to be solved;

[0007] Step S2: Optimize the structure of the original Harris hawk algorithm. The specific optimization measures are as follows: First, cancel the steps of centrally calculating the objective function value in the original Harris hawk algorithm except for the step of initializing the population, but record the results while the algorithm obtains a better solution to save computational time. Second, introduce a random unit replacement mechanism at the position before the end of each iteration optimization in the original Harris hawk algorithm to enhance the global search performance. Third, delete the search steps in the original Harris hawk algorithm where the absolute value of the energy factor E is greater than or equal to 1 to reduce the influence on the optimization effect of the random unit replacement mechanism. The optimized Harris hawk algorithm is called the Harris hawk algorithm based on random unit replacement, and use this Harris hawk algorithm based on random unit replacement to solve the key parameters to obtain the key parameters for optimizing the weight of the disc spring.

[0008] The specific process of determining the objective function for optimizing the weight of the disc spring and the key parameters to be solved in step S1 is as follows:

[0009] S1.1: Determine the objective function for optimizing the weight of the disc spring. This objective function is shown in Equation (1):

[0010]

[0011] where F(x) is the objective function, representing the weight of the disc spring, with the measurement unit of pound (lb), ρ represents the density of the disc spring material, with the measurement unit of pound per cubic inch (lb / in 3 ), V BS is the volume of the disc spring, with the measurement unit of cubic inch (in 3 ), π is the pi, D ot is the outer diameter of the disc spring, with the measurement unit of inch (in), D inn is the inner diameter of the disc spring, with the measurement unit of inch (in), t s is the thickness of the disc spring, with the measurement unit of inch (in);

[0012] S1.2: Determine the constraint conditions for optimizing the weight of the disc spring. The constraint conditions are shown in Equations (2) to (8):

[0013]

[0014]

[0015] g 3 (x) = δ l - δ max ≥0 (4)

[0016] g 4 (x) = H - h - ts ≥ 0 (5)

[0017] g 5 (x) = D mmax -D ot ≥ 0 (6)

[0018] g 6 (x) = D ot -D inn ≥ 0 (7)

[0019]

[0020] wherein, g 1 (x) is the stress constraint generated by the radial shortening of the conical spring, g 2 (x) is the stiffness constraint of the conical spring, g 3 (x) is the defined deflection constraint of the conical spring, g 4 (x) is the relationship constraint between the thickness and height of the conical spring, g 5 (x) is the outer diameter constraint of the conical spring, g 6 (x) is the relationship constraint between the outer diameter and inner diameter of the conical spring, g 7 (x) is the geometric dimension constraint of the conical spring, h is the height of the conical spring, the measurement unit is inch (in), S is the allowable strength of the conical spring, the measurement unit is kilopound-force per square inch (kpsi), E is the elastic modulus of the conical spring, the measurement unit is pound-force per square inch (psi), δ max is the maximum deflection of the conical spring, the measurement unit is inch (in), μ is the Poisson's coefficient of the conical spring material, P max is the maximum load of the conical spring, the measurement unit is pound (lb), H is the maximum limit of the height of the conical spring, the measurement unit is inch (in), D max is the maximum outer diameter of the conical spring, the measurement unit is inch (in), δ l is the defined deflection, δ l = f(a)h, represents the ratio of the height to the thickness of the conical spring, f(a) represents the load deformation characteristic of the conical spring, the measurement unit is inch (in), let K = D ot / D inn , α, β, γ are temporary variables, α, β, γ are respectively calculated from Equations (9) to (11):

[0021]

[0022]

[0023]

[0024] S1.3. Determine the parameter values in the constraint conditions shown in Equations (2) to (8), wherein,ρ = 0.283 lb / in 3 , S = 200 kpsi, E = 30×10 6 psi, δm ax = 0.2 in, μ = 0.3, Pm ax = 5400 lb, H = 2 in, D max = 12.01 in, the relationship between a and the load deformation characteristic f(a) is as follows: when a < 1.45, f(a) = 1; when 1.45 ≤ a < 1.55, f(a) = 0.85; when 1.55 ≤ a < 1.65, f(a) = 0.77; when 1.65 ≤ a < 1.75, f(a) = 0.71; when 1.75 ≤ a < 1.85, f(a) = 0.66; when 1.85 ≤ a < 1.95, f(a) = 0.63; when 1.95 ≤ a < 2.05, f(a) = 0.6; when 2.05 ≤ a < 2.15, f(a) = 0.58; when 2.15 ≤ a < 2.25, f(a) = 0.56; when 2.25 ≤ a < 2.35, f(a) = 0.55; when 2.35 ≤ a < 2.45, f(a) = 0.53; when 2.45 ≤ a < 2.55, f(a) = 0.52; when 2.55 ≤ a < 2.65, f(a) = 0.51; when 2.65 ≤ a < 2.75, f(a) = 0.51; when a ≥ 2.75, f(a) = 0.50. The remaining four parameters are the key parameters to be solved, namely the outer diameter D ot of the spring, the inner diameter D inn of the spring, the thickness t s of the spring and the height h of the spring. Their parameter ranges are: 5 in ≤ D ot ≤ 15 in, 5 in ≤ D inn ≤ 15 in, 0.01 in ≤ t s ≤ 6 in, 0.05 in ≤ h ≤ 0.5 in. Let the vector X represent the key parameters to be solved, the vector LB represent the lower bound of X, and the vector UB represent the upper bound of X. Their expressions are shown by equations (12) to (14) respectively.

[0025] X = [D ot , D inn , t s , h] (12)

[0026] LB = [5, 5, 0.01, 0.05] (13)

[0027] UB = [15, 15, 6, 0.5] (14)

[0028] Among them, the first-dimensional data of LB represents the lower limit of D ot of the spring, and the second-dimensional data represents Dinn The lower limit, the third - dimensional data represents t s The lower limit, the fourth - dimensional data represents the lower limit of h, the first - dimensional data of UB represents D ot The upper limit, the second - dimensional data represents D inn The upper limit, the third - dimensional data represents t s The upper limit, the fourth - dimensional data represents the upper limit of h;

[0029] S1.4. Use the vector X to transform the objective function shown in formula (1) to obtain the final objective function represented by formula (15) as:

[0030] F(X)=0.07075π((X 1 ) 2 -(X 2 ) 2 )X 3 (15)

[0031] Wherein, X 1 represents the first - dimensional data of the vector X, X 2 represents the second - dimensional data of the vector X, X 3 represents the third - dimensional data of the vector X, (·) 2 represents the operation of squaring the data.

[0032] In the step S2, the Harris hawk algorithm based on random unit permutation is used to solve the key parameters, and the specific process of obtaining the key parameters for the weight optimization of the disc spring is as follows:

[0033] S2.1. Initialize the population to obtain the initial population: Corresponding the vector X to the individuals of the population, the individual dimension D is 4, the first - dimensional data of the individual corresponds to D ot , the second - dimensional data of the individual corresponds to D inn , the third - dimensional data of the individual corresponds to t s , the fourth - dimensional data of the individual corresponds to h; Set the population size N of the Harris hawk algorithm based on random unit permutation to 30, randomly initialize 30 individuals to obtain the initial population, and the initialization method is shown by formula (16):

[0034]

[0035] Wherein, represents the lower limit of the j - th dimensional data of the i - th individual, represents the j - th dimensional data of the i - th individual generated by the random function, and its range is [0, 1], ° represents the Hadamard product operator of the matrix, that is, multiplying the elements in the same position of the matrix, UB ii =UB, LB ii =LB, X iiRepresents the \(i\)-th individual of the population, Represents the \(j\)-th dimensional data of the \(i\)-th individual, where \(i = 1, 2, \ldots, 30\) and \(j = 1, 2, 3, 4\);

[0036] S2.2. Evaluate the initial population. The specific process is as follows: According to the objective function shown in Equation (15), calculate the objective function value of each individual in the initial population, and judge each individual in the initial population according to the constraint conditions in Equations (2) to (8). If a certain individual currently cannot satisfy all the constraint conditions, then update this individual, and re-randomly assign each dimensional data of this individual within the range of the upper and lower limits corresponding to this dimensional data once. After this individual is updated currently, instead of calculating its objective function value through Equation (15), directly set its objective function value to 10 10 , if a certain individual can currently satisfy all the constraint conditions, then this individual remains unchanged. At this time, the 0th generation population is obtained. Denote the individual with the smallest objective function value among the individuals in the 0th generation population that can satisfy all the constraint conditions as gBest, and denote the objective function value of individual gBest as F(gBest). Set the global optimal individual of the \(i\)-th individual and denote it as pBest ii , and adopt the \(X\) in the 0th generation population ii Initialize pBest ii The value of, and denote the objective function value of pBest ii as F(pBest ii );

[0037] S2.3. Set the iteration number variable \(t\) and the maximum iteration number \(T\). Initialize the iteration number \(t\) to 1 and set the maximum iteration number \(T\) to 1000;

[0038] S2.4. Conduct the \(t\)-th iteration on the population to obtain the \(t\)-th generation population. The specific process is as follows:

[0039] S2.4.1. Set the individual number \(i\) and initialize the individual number \(i\) to 1;

[0040] S2.4.2. Update the \(i\)-th individual to obtain the \(i\)-th individual \(X\) i (t) of the \(t\)-th generation population. The specific process is as follows:

[0041] S2.4.2.1. Set the energy factor of the \(i\)-th individual in the \(t\)-th iteration as Calculate the energy factor of the \(i\)-th individual in the \(t\)-th iteration according to Equation (17) This parameter is used for the switching of the algorithm search mode:

[0042]

[0043] Among them, represents the random number of the i-th individual in the t-th iteration, which is generated by a random function, and

[0044] S2.4.2.2. If then go to step S2.4.2.3; if then first let the value of X i (t) be X i (t - 1), and then go to step S2.4.2.4. X i (t - 1) is the i-th individual of the (t - 1)-th generation population, and | | is the absolute value symbol;

[0045] S2.4.2.3. Generate a random number using the random number function and If and update the i-th individual according to the formula shown in Equation (18) to obtain X i (t); if and update the i-th individual according to the formula shown in Equation (23) to obtain X i (t); if and update the i-th individual according to the formula shown in Equation (24) to obtain X i (t); if and update the i-th individual according to the formula shown in Equation (28) to obtain X i (t); after obtaining X i (t) using Equation (18), (23), (24), or (28), calculate the objective function value of X i (t) according to Equation (15). If the objective function value of X i (t) is less than or equal to the objective function value of X i (t - 1), then keep X i (t) unchanged. If the objective function value of X i (t) is greater than the objective function value of X i (t - 1), then update X i (t) to be equal to X i (t - 1); substitute the data of X i (t) into Equations (2) to (8) respectively, and judge whether X i (t) simultaneously satisfies the constraint conditions of Equations (2) to (8). If it satisfies simultaneously, then keep X i (t) unchanged. Otherwise, update X i (t) again, and set X iEach dimension data of (t) is randomly reassigned once within the range of the upper and lower limits corresponding to the data in that dimension, X i After the current update of (t), instead of calculating its objective function value F(X i (t)) through formula (15), its objective function value F(X i (t)) is directly set to 10 10 ;

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057] Among them, X i (t) represents the i-th individual of the t-th generation population, represents the four intermediate individuals generated for the i-th individual at the t-th iteration. | | represents the absolute value symbol, and the value of θ is 1.5, represents the first 1×D-dimensional random number vector generated for the i-th individual at the t-th iteration. Each dimension data of this random number vector has a lower limit of 0 and an upper limit of 1, represents the second 1×D-dimensional random number vector generated for the i-th individual at the t-th iteration. Each dimension data of this random number vector has a lower limit of 0 and an upper limit of 1, represents the third 1×D-dimensional random number vector generated for the i-th individual at the t-th iteration. Each dimension data of this random number vector has a lower limit of 0 and an upper limit of 1, represents the random number generated for the i-th individual at the t-th iteration, which is used to perturb the current optimal individual to enhance the population diversity. LF() is the Lévy flight function, represents X when solving the i-th individual at the t-th iteration 1(t-1), X 2 (t-1),...,X N The average value of (t-1) is given by equation (26), where Γ is the gamma function, Indicates a D-dimensional random number vector with one row generated for the i-th individual in the t-th iteration. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. If X i The objective function value F(X i (t)) is set equal to 10 10 , then directly obtain X i The objective function value F(X i (t)), if X i The objective function value F(X i (t)) is not set equal to 10 10 , then according to formula (15) calculate its objective function value F(X i (t)), and then make the following judgment: If F(X i (t))<F(gBest), then use X i (t) Update the variable value of gBest, otherwise do not update the variable value of gBest; if F(X i (t))<F(pBest i ), then use X i (t) Update pBest i Otherwise, pBest is not updated. i Variable value; so far, X i (t) Update completed, go to step S2.4.2.4;

[0058] S2.4.2.4. Set four intermediate individuals and According to formula (29) and formula (30), calculate the two intermediate individuals and

[0059]

[0060]

[0061] in, A D-dimensional random number vector with one row generated by a random function. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. is a D-dimensional random number vector with one row generated by a random function. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. ~ is a bitwise negation operator. are any four different integers in the range [1, N] except i;

[0062] Substitute and into equations (31) and (32) respectively to obtain two intermediate individuals and

[0063]

[0064]

[0065] where is a 1-row D-dimensional random number vector generated by a random function. The lower limit of each dimension data of this random number vector is 0, and the upper limit is 1. is a 1-row D-dimensional random number vector generated by a random function. The lower limit of each dimension data of this random number vector is 0, and the upper limit is 1. is a random number generated by a random function, and is a random number generated by a random function, and e is the Napier constant, with a value of 2.718281828459045;

[0066] Substitute the data of Branch into equations (2) to (8) respectively to determine whether and simultaneously satisfy the constraint conditions of equations (2) to (8). If and any one of them does not satisfy simultaneously, update the intermediate individual that does not satisfy simultaneously once, randomly re-assign each dimension data within the range of the upper and lower limits corresponding to that dimension data, and directly set its objective function value to 10 10 , and then substitute it into equation (33) to obtain the intermediate individual If and both satisfy, then substitute and directly into equation (33) to obtain the intermediate individual

[0067]

[0068] S2.4.2.5. Use to update X i (t), calculate the objective function value of X i (t) using equation (15). If the objective function value of X i (t) is less than the objective function value of gBest, then use X i(t) Update the value of gBest, otherwise do not update gBest. If the objective function value of X i (t) is less than the objective function value of pBest i , then use X i (t) to update pBest i , otherwise do not update pBest i ;

[0069] S2.4.3. Determine whether the current value of i is equal to N. If not, update the value of i with the sum of the current value of i plus 1, and then return to step S2.4.2 to update the next individual. If it is equal to N, the t-th iteration is completed, and N individuals X 1 (t) to X N (t) of the t-th generation population are obtained. At this time, enter the next step;

[0070] S2.4.4. Determine whether the current value of t is equal to T. If not, update the value of t with the sum of the current value of t plus 1, and then return to step S2.4 for the next iteration. If it is equal to T, enter the next step;

[0071] S2.5. Output the currently obtained gBest. The currently obtained gBest is the key parameter for solving the optimized weight of the disc spring.

[0072] Compared with the prior art, the advantage of the present invention is that after determining the objective function of disc spring weight optimization and the key parameters to be solved, the original Harris Hawk algorithm structure is optimized, and the specific optimization measures are: first, the steps of the original Harris Hawk algorithm except for the initialization population step for centrally calculating the objective function value are cancelled, and the results are recorded while the algorithm obtains a better solution to save calculation time; second, a random unit replacement mechanism is introduced before the end of each iteration of the original Harris Hawk algorithm to enhance the global search performance; third, the search steps in which the absolute value of the energy factor E of the original Harris Hawk algorithm is greater than or equal to 1 are deleted to reduce the impact on the optimization effect of the random unit replacement mechanism; the optimized Harris Hawk algorithm is called the Harris Hawk algorithm based on random unit replacement, and The Harris Eagle algorithm based on random unit replacement is used to solve key parameters and obtain key parameters for optimizing the weight of disc springs. In the present invention, the Harris Eagle algorithm based on random unit replacement does not care about the specific mathematical characteristics of the problem to be solved, and can effectively avoid falling into local extreme values. The Harris Eagle algorithm based on random unit replacement limits the operation of the algorithm by the number of iterations, and its solution time is controllable to avoid excessive calculation time. The random unit replacement mechanism of the Harris Eagle algorithm based on random unit replacement is used to improve population diversity, and its solution speed and solution accuracy are further improved. In addition, the present invention optimizes the structure while using the original Harris Eagle algorithm to ensure that the introduced mechanism plays the best effect. Therefore, the present invention has fast convergence speed and high solution accuracy, can further optimize the weight of disc springs, and has high application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 This is a simulation result diagram of the disc spring weight optimization method based on the random unit replacement Harris Eagle algorithm of the present invention. DETAILED DESCRIPTION

[0074] The present invention is further described in detail below with reference to the accompanying drawings.

[0075] Embodiment: A disc spring weight optimization method based on random unit permutation Harris Eagle algorithm comprises the following steps:

[0076] Step S1, determining the objective function of disc spring weight optimization and the key parameters to be solved;

[0077] Step S2: Optimize the original Harris hawk algorithm structure. The specific optimization measures are as follows: First, cancel the steps of centrally calculating the objective function value in the original Harris hawk algorithm except for the step of initializing the population. Instead, record the results while the algorithm obtains a better solution to save computational time. Second, introduce a random unit permutation mechanism at the position before the end of each iteration optimization in the original Harris hawk algorithm to enhance the global search performance. Third, delete the search steps in the original Harris hawk algorithm where the absolute value of the energy factor E is greater than or equal to 1 to reduce the impact on the optimization effect of the random unit permutation mechanism. The optimized Harris hawk algorithm is called the Harris hawk algorithm based on random unit permutation, and use this Harris hawk algorithm based on random unit permutation to solve the key parameters to obtain the key parameters for optimizing the disc spring weight.

[0078] In this embodiment, the specific process of determining the objective function for optimizing the disc spring weight and the key parameters to be solved in step S1 is as follows:

[0079] S1.1: Determine the objective function for optimizing the disc spring weight. The objective function is shown in Equation (1):

[0080]

[0081] Among them, F(x) is the objective function, representing the weight of the disc spring, with the measurement unit of pounds (lb), ρ represents the density of the disc spring material, with the measurement unit of pounds per cubic inch (lb / in 3 ), V BS is the volume of the disc spring, with the measurement unit of cubic inches (in 3 ), π is the pi, D ot is the outer diameter of the disc spring, with the measurement unit of inches (in), D inn is the inner diameter of the disc spring, with the measurement unit of inches (in), t s is the thickness of the disc spring, with the measurement unit of inches (in);

[0082] S1.2: Determine the constraint conditions for optimizing the disc spring weight. The constraint conditions are shown in Equations (2) to (8):

[0083]

[0084]

[0085] g 3 (x) = δ l -δ max ≥0 (4)

[0086] g 4 (x) = H - h - t s ≥0 (5)

[0087] g 5g(x) = D max -D ot ≥0 (6)

[0088] g 6 g(x) = D ot -D inn ≥0 (7)

[0089]

[0090] where g 1 g(x) is the stress constraint generated by the radial shortening of the conical spring, g 2 g(x) is the stiffness constraint of the conical spring, g 3 g(x) is the limited deflection constraint of the conical spring, g 4 g(x) is the relationship constraint between the thickness and height of the conical spring, g 5 g(x) is the outer diameter constraint of the conical spring, g 6 g(x) is the relationship constraint between the outer diameter and inner diameter of the conical spring, g 7 g(x) is the geometric dimension constraint of the conical spring, h is the height of the conical spring, the measurement unit is inch (in), S is the allowable strength of the conical spring, the measurement unit is kips per square inch (kpsi), E is the elastic modulus of the conical spring, the measurement unit is pounds per square inch (psi), δ max is the maximum deflection of the conical spring, the measurement unit is inch (in), μ is the Poisson's coefficient of the conical spring material, P max is the maximum load of the conical spring, the measurement unit is pound (lb), H is the maximum limit of the height of the conical spring, the measurement unit is inch (in), D max is the maximum outer diameter of the conical spring, the measurement unit is inch (in), δ l is the limited deflection, δ l = f(a)h, represents the ratio of the height to the thickness of the conical spring, f(a) represents the load deformation characteristic of the conical spring, the measurement unit is inch (in), let K = D ot / D inn , α, β, γ are temporary variables, and α, β, γ are calculated by equations (9) to (11) respectively:

[0091]

[0092]

[0093]

[0094] S1.3. Determine the parameter values in the constraint conditions shown in equations (2) to (8), where ρ = 0.283 lb / in 3 , S = 200 kpsi, E = 30×10 6 psi, δmax = 0.2 in, μ = 0.3, P max = 5400 lb, H = 2 in, D max = 12.01 in, the relationship between a and the load deformation characteristic f(a) is as follows: when a < 1.45, f(a) = 1; when 1.45 ≤ a < 1.55, f(a) = 0.85; when 1.55 ≤ a < 1.65, f(a) = 0.77; when 1.65 ≤ a < 1.75, f(a) = 0.71; when 1.75 ≤ a < 1.85, f(a) = 0.66; when 1.85 ≤ a < 1.95, f(a) = 0.63; when 1.95 ≤ a < 2.05, f(a) = 0.6; when 2.05 ≤ a < 2.15, f(a) = 0.58; when 2.15 ≤ a < 2.25, f(a) = 0.56; when 2.25 ≤ a < 2.35, f(a) = 0.55; when 2.35 ≤ a < 2.45, f(a) = 0.53; when 2.45 ≤ a < 2.55, f(a) = 0.52; when 2.55 ≤ a < 2.65, f(a) = 0.51; when 2.65 ≤ a < 2.75, f(a) = 0.51; when a ≥ 2.75, f(a) = 0.50. The remaining four parameters are the key parameters to be solved, namely the outer diameter D ot of the spring, the inner diameter D inn of the spring, the thickness t s of the spring and the height h of the spring. Their parameter ranges are: 5 in ≤ D ot ≤ 15 in, 5 in ≤ D inn ≤ 15 in, 0.01 in ≤ t s ≤ 6 in, 0.05 in ≤ h ≤ 0.5 in. Let the vector X represent the key parameters to be solved, the vector LB represent the lower bound of X, and the vector UB represent the upper bound of X. Their expressions are shown by equations (12) to (14) respectively.

[0095] X = [D ot , D in , t s , h] (12)

[0096] LB = [5, 5, 0.01, 0.05] (13)

[0097] UB = [15, 15, 6, 0.5] (14)

[0098] Among them, the first-dimensional data of LB represents the lower limit of D ot , the second-dimensional data represents the lower limit of D inn , the third-dimensional data represents the lower limit of t s , and the fourth-dimensional data represents the lower limit of h. The first-dimensional data of UB represents the upper limit of D otThe upper limit, the second-dimensional data represents D inn The upper limit, the third-dimensional data represents t s The upper limit, the fourth-dimensional data represents the upper limit of h;

[0099] S1.4. Use the vector X to transform the objective function shown in formula (1) to obtain the final objective function represented by formula (15) as:

[0100] F(X) = 0.07075π((X 1 ) 2 -(X 2 ) 2 )X 3 (15)

[0101] Among them, X 1 represents the first-dimensional data of the vector X, X 2 represents the second-dimensional data of the vector X, X 3 represents the third-dimensional data of the vector X, (·) 2 represents the square operation on the data.

[0102] In step S2, the Harris hawk algorithm based on random unit permutation is used to solve the key parameters. The specific process of obtaining the key parameters for the optimization of the disc spring weight is as follows:

[0103] S2.1. Initialize the population to obtain the initial population: Corresponding the vector X to the individuals of the population, the individual dimension D is 4. The first-dimensional data of the individual corresponds to D ot , the second-dimensional data of the individual corresponds to D inn , the third-dimensional data of the individual corresponds to t s , the fourth-dimensional data of the individual corresponds to h; Set the population size N of the Harris hawk algorithm based on random unit permutation to 30, and randomly initialize 30 individuals to obtain the initial population. The initialization method is shown by formula (16):

[0104]

[0105] Among them, represents the lower limit of the j-th dimensional data of the i-th individual, represents the j-th dimensional data of the i-th individual generated by the random function, and its range is [0, 1], represents the Hadamard product operator of the matrix, that is, multiplying the elements in the same position of the matrix, UB ii = UB, LB ii = LB, X ii represents the i-th individual of the population, Denote the data of the \(i\)-th individual in the \(j\)-th dimension, where \(i = 1, 2, \ldots, 30\) and \(j = 1, 2, 3, 4\);

[0106] S2.2. Evaluate the initial population. The specific process is as follows: According to the objective function shown in Equation (15), calculate the objective function value of each individual in the initial population, and determine each individual in the initial population according to the constraint conditions in Equations (2) to (8). If a current individual cannot satisfy all the constraint conditions simultaneously, update the individual by randomly re-assigning each dimension data of the individual within the range of the upper and lower limits corresponding to that dimension data once. After the individual is updated currently, instead of calculating its objective function value through Equation (15), directly set its objective function value to 10 10 , if a current individual can satisfy all the constraint conditions simultaneously, the individual remains unchanged. At this time, the 0th generation population is obtained. Denote the individual with the smallest objective function value among the individuals in the 0th generation population that can satisfy all the constraint conditions as gBest, and denote the objective function value of the individual gBest as F(gBest). Set the globally optimal individual of the \(i\)-th individual and denote it as pBest ii , and adopt \(X\) in the 0th generation population ii Initialize the value of pBest ii , and denote the objective function value of pBest ii as F(pBest ii );

[0107] S2.3. Set the iteration number variable \(t\) and the maximum iteration number \(T\). Initialize the iteration number \(t\) to 1 and set the maximum iteration number \(T\) to 1000;

[0108] S2.4. Conduct the \(t\)-th iteration on the population to obtain the \(t\)-th generation population. The specific process is as follows:

[0109] S2.4.1. Set the individual number \(i\) and initialize the individual number \(i\) to 1;

[0110] S2.4.2. Update the \(i\)-th individual to obtain the \(i\)-th individual \(X\) i (t) of the \(t\)-th generation population. The specific process is as follows:

[0111] S2.4.2.1. Set the energy factor of the \(i\)-th individual in the \(t\)-th iteration as Calculate the energy factor of the \(i\)-th individual in the \(t\)-th iteration according to Equation (17) This parameter is used for the switching of the algorithm search mode:

[0112]

[0113] where, denotes the random number of the \(i\)-th individual in the \(t\)-th iteration, which is generated by a random function, and

[0114] S2.4.2.2. If then go to step S2.4.2.3; if then first set the value of X i (t) to be X i (t - 1), and then go to step S2.4.2.4. X i (t - 1) is the i-th individual of the (t - 1)-th generation population, and || is the absolute value symbol;

[0115] S2.4.2.3. Generate a random number using the random number function and If and update the i-th individual according to the formula shown in Equation (18) to obtain X i (t); if and update the i-th individual according to the formula shown in Equation (23) to obtain X i (t); if and update the i-th individual according to the formula shown in Equation (24) to obtain X i (t); if and update the i-th individual according to the formula shown in Equation (28) to obtain X i (t); after obtaining X i (t) using Equation (18), (23), (24), or (28), calculate the objective function value of X i (t) according to Equation (15). If the objective function value of X i (t) is less than or equal to the objective function value of X i (t - 1), then keep X i (t) unchanged. If the objective function value of X i (t) is greater than the objective function value of X i (t - 1), then update X i (t) to be equal to X i (t - 1); substitute the data of X i (t) into Equations (2) to (8) respectively to judge whether X i (t) simultaneously satisfies the constraint conditions of Equations (2) to (8). If it satisfies simultaneously, then keep X i (t) unchanged. Otherwise, update X i (t) again, and randomly reassign each dimension data of X i (t) within the range of the upper and lower limits corresponding to this dimension data. X i(t) After the current update, instead of calculating its objective function value F(X i (t)) using Equation (15), its objective function value F(X i (t)) is directly set to 10 10 ;

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122]

[0123]

[0124]

[0125]

[0126]

[0127] Among them, X i (t) represents the i-th individual of the t-th generation population, represents the four intermediate individuals generated for the i-th individual at the t-th iteration, || represents the absolute value symbol, and the value of θ is 1.5, represents the first 1×D random number vector generated for the i-th individual at the t-th iteration. Each dimension data of this random number vector has a lower limit of 0 and an upper limit of 1, represents the second 1×D random number vector generated for the i-th individual at the t-th iteration. Each dimension data of this random number vector has a lower limit of 0 and an upper limit of 1, represents the third 1×D random number vector generated for the i-th individual at the t-th iteration. Each dimension data of this random number vector has a lower limit of 0 and an upper limit of 1, represents the random number generated for the i-th individual at the t-th iteration, which is used to perturb the current optimal individual to improve population diversity. LF() is the Lévy flight function, represents X 1 (t - 1), X 2 (t - 1),..., X NThe average value of (t-1) is given by equation (26), where Γ is the gamma function, Indicates a D-dimensional random number vector with one row generated for the i-th individual in the t-th iteration. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. If X i The objective function value F(X i (t)) is set equal to 10 10 , then directly obtain X i The objective function value F(X i (t)), if X i The objective function value F(X i (t)) is not set equal to 10 10 , then according to formula (15) calculate its objective function value F(X i (t)), and then make the following judgment: If F(X i (t))<F(gBest), then use X i (t) Update the variable value of gBest, otherwise do not update the variable value of gBest; if F(X i (t))<F(pBest i ), then use X i (t) Update pBest i Otherwise, pBest is not updated. i Variable value; so far, X i (t) Update completed, go to step S2.4.2.4;

[0128] S2.4.2.4. Set four intermediate individuals and According to formula (29) and formula (30), calculate the two intermediate individuals and

[0129]

[0130]

[0131] in, A D-dimensional random number vector with one row generated by a random function. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. is a D-dimensional random number vector with one row generated by a random function. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. ~ is a bitwise negation operator. are any four different integers in the range [1, N] except i;

[0132] Will and Substitute into Equation (31) and Equation (32) respectively to obtain two intermediate individuals and

[0133]

[0134]

[0135] wherein, is a 1-row D-dimensional random number vector generated by a random function, and the lower limit of each dimension data of this random number vector is 0 and the upper limit is 1; is a 1-row D-dimensional random number vector generated by a random function, and the lower limit of each dimension data of this random number vector is 0 and the upper limit is 1; is a random number generated by a random function, and is a random number generated by a random function, and e is the Napier constant, with a value of 2.718281828459045;

[0136] Substitute the data of and into Equation (2) to Equation (8) respectively, and judge whether Branch simultaneously satisfies the constraint conditions of Equation (2) to Equation (8). If and do not satisfy simultaneously, update the intermediate individual that does not satisfy simultaneously once, re-randomly assign each dimension data within the range of the upper and lower limits corresponding to this dimension data, and directly set its objective function value to 10 10 , and then substitute it into Equation (33) to obtain the intermediate individual If and both satisfy, then directly substitute and into Equation (33) to obtain the intermediate individual

[0137]

[0138] S2.4.2.5. Use to update X i (t), calculate the objective function value of X i (t) using Equation (15). If the objective function value of X i (t) is less than the objective function value of gBest, then update the value of gBest using X i (t), otherwise do not update gBest. If the objective function value of X i (t) is less than pBest iIf it is the objective function value, then use X i (t) to update pBest i , otherwise do not update pBest i ;

[0139] S2.4.3. Determine whether the current value of i is equal to N. If not, update the value of i with the sum of the current value of i plus 1, and then return to step S2.4.2 to update the next individual. If it is equal to N, the t-th iteration is completed, and N individuals X 1 (t) to X N (t) of the t-th generation population are obtained, and then enter the next step;

[0140] S2.4.4. Determine whether the current value of t is equal to T. If not, update the value of t with the sum of the current value of t plus 1, and then return to step S2.4 for the next iteration. If it is equal to T, enter the next step;

[0141] S2.5. Output the currently obtained gBest. The currently obtained gBest is the key parameter for the optimized weight of the disc spring obtained by the solution.

[0142] The simulation result diagram of the method for optimizing the weight of the disc spring based on the random unit permutation Harris hawk algorithm of the present invention is as Figure 1 shown. Analysis Figure 1 shows that the method for optimizing the weight of the disc spring based on the random unit permutation Harris hawk algorithm of the present invention has a fast convergence speed and high optimization accuracy. It can obtain the key parameters for the optimized weight of the disc spring under the condition of satisfying all constraint conditions, so that the optimization effect of the disc spring weight is significantly improved compared with the original method; the random unit permutation mechanism adopted by the present invention can enhance the diversity of the population, and then increase the probability of discovering the optimal search space. The method proposed by the present invention can reliably solve the problem of optimizing the weight of the disc spring.

Claims

1. A method for optimizing the weight of disc springs based on a Harris hawk algorithm with random unit permutation, characterized in that it includes the following steps: Step S1, determine the objective function for optimizing the weight of the disc spring and the key parameters to be solved; Step S2, optimize the structure of the original Harris hawk algorithm, and call the optimized Harris hawk algorithm the Harris hawk algorithm based on random unit permutation. Use this Harris hawk algorithm based on random unit permutation to solve the key parameters and obtain the key parameters for optimizing the weight of the disc spring; The specific process of determining the objective function for optimizing the weight of the disc spring and the key parameters to be solved in the said step S1 is as follows: S1.1, determine the objective function for optimizing the weight of the disc spring, and this objective function is shown in formula (1): Where F(x) is the objective function, which represents the weight of the disc spring in pounds (lb), and ρ represents the density of the disc spring material in pounds / cubic inch (lb / in 3 ), V BS The volume of the disc spring is measured in cubic inches (in 3 ), π is the circumference of a circle, D ot D is the outer diameter of the disc spring, measured in inches (in). inn is the inner diameter of the disc spring, measured in inches (in), t s is the disc spring thickness, measured in inches (in); S1.2, determine the constraint conditions for optimizing the weight of the disc spring, and the constraint conditions are shown in formulas (2) to (8): g 3 f(x) = δ l -δ max ≥ 0 (4) g 4 g(x) = H - h - t s ≥ 0 (5) g 5 f(x) = D max -D ot ≥ 0 (6) g 6 f(x) = D ot -D inn ≥ 0 (7) Among them, g 1 (x) is the stress constraint generated by the radial shortening of the disc spring, g 2 (x) is the stiffness constraint of the disc spring, g 3 (x) is the limited deflection constraint of the disc spring, g 4 (x) is the relationship constraint between the thickness and height of the disc spring, g 5 (x) is the outer diameter constraint of the disc spring, g 6 (x) is the relationship constraint between the outer diameter and inner diameter of the disc spring, g 7 (x) is the geometric dimension constraint of the disc spring, h is the height of the disc spring, and the measurement unit is inch (in), S is the allowable strength of the disc spring, and the measurement unit is kilopound force per square inch (kpsi), E is the elastic modulus of the disc spring, and the measurement unit is pound force per square inch (psi), δ max is the maximum deflection of the disc spring, and the measurement unit is inch (in), μ is the Poisson's coefficient of the disc spring material, P max is the maximum load of the disc spring, and the measurement unit is pound (lb), H is the maximum limit of the disc spring height, and the measurement unit is inch (in), D max is the maximum outer diameter of the disc spring, and the measurement unit is inch (in), δ l is the limited deflection, δ l = f(a)h, represents the ratio of the disc spring height to the thickness, f(a) represents the load deformation characteristic of the disc spring, and the measurement unit is inch (in). Let K = D ot / D inn , α, β, γ are temporary variables, and α, β, γ are calculated from equations (9) to (11) respectively: S1.

3. Determine the parameter values in the constraint conditions shown in formulas (2) to (8), where ρ = 0.283 lb / in 3 , S = 200 kpsi, E = 30×10 6 psi, δ max = 0.2 in, μ = 0.3, P max = 5400 lb, H = 2 in, D max = 12.01 in, the relationship between a and the load deformation characteristic f(a) is as follows: when a < 1.45, f(a) = 1; when 1.45 ≤ a < 1.55, f(a) = 0.85; when 1.55 ≤ a < 1.65, f(a) = 0.77; when 1.65 ≤ a < 1.75, f(a) = 0.71; when 1.75 ≤ a < 1.85, f(a) = 0.66; when 1.85 ≤ a < 1.95, f(a) = 0.63; when 1.95 ≤ a < 2.05, f(a) = 0.6; when 2.05 ≤ a < 2.15, f(a) = 0.58; when 2.15 ≤ a < 2.25, f(a) = 0.56; when 2.25 ≤ a < 2.35, f(a) = 0.55; when 2.35 ≤ a < 2.45, f(a) = 0.53; when 2.45 ≤ a < 2.55, f(a) = 0.52; when 2.55 ≤ a < 2.65, f(a) = 0.51; when 2.65 ≤ a < 2.75, f(a) = 0.51; when a ≥ 2.75, f(a) = 0.

50. The remaining four parameters are the key parameters to be solved, namely the outer diameter D ot of the spring, the inner diameter D inn of the spring, the thickness t s of the spring and the height h of the spring. Their parameter ranges are: 5 in ≤ D ot ≤ 15 in, 5 in ≤ D inn ≤ 15 in, 0.01 in ≤ t s ≤ 6 in, 0.05 in ≤ h ≤ 0.5 in. Use the vector X to represent the key parameters to be solved, use the vector LB to represent the lower bound of X, and use the vector UB to represent the upper bound of X. Their expressions are shown in formulas (12) to (14) respectively; X = [D ot , D inn , t s , h] (12) LB = [5, 5, 0.01, 0.05] (13) UB = [15, 15, 6, 0.5] (14) Among them, the first-dimensional data of LB represents the lower limit of D ot The lower limit of, the second-dimensional data represents D inn The lower limit of, the third-dimensional data represents t s The lower limit of, the fourth-dimensional data represents the lower limit of h. The first-dimensional data of UB represents D ot The upper limit of, the second-dimensional data represents D inn The upper limit of, the third-dimensional data represents t s The upper limit of, the fourth-dimensional data represents the upper limit of h; S1.4, use the vector X to transform the objective function shown in formula (1), and the final objective function is expressed by formula (15) as: F(X) = 0.07075π((X 1 ) 2 -(X 2 ) 2 )X 3 (15) Among them, X 1 represents the first-dimensional data of vector X, X 2 represents the second-dimensional data of vector X, X 3 represents the third-dimensional data of vector X, (·) 2 represents performing a square operation on the data; The specific process of using the Harris hawk algorithm based on random unit permutation to solve the key parameters and obtain the key parameters for optimizing the weight of the disc spring in the said step S2 is as follows: S2.

1. Initialize the population to obtain the initial population: Corresponding the vector X to the individuals of the population, the individual dimension D is 4. The first-dimensional data of the individual corresponds to D ot , the second-dimensional data of the individual corresponds to D inn , the third-dimensional data of the individual corresponds to t s , and the fourth-dimensional data of the individual corresponds to h; Set the population size N of the Harris hawk algorithm based on random unit permutation to 30, and randomly initialize 30 individuals to obtain the initial population. The initialization method is shown in Equation (16): Among them, represents the lower limit of the j-th dimension data of the i-th individual, represents the j-th dimension data of the i-th individual generated by a random function, and its range is [0, 1], represents the Hadamard product operator of matrices, that is, multiplying the elements in the same position of the matrices, UB ii = UB, LB ii = LB, X ii represents the i-th individual of the population, represents the j-th dimension data of the i-th individual, i = 1, 2,..., 30, j = 1, 2, 3, 4; S2.

2. Evaluate the initial population. The specific process is as follows: According to the objective function shown in Equation (15), calculate the objective function value of each individual in the initial population, and judge each individual in the initial population according to the constraint conditions in Equations (2) to (8). If a current individual cannot satisfy all the constraint conditions at the same time, update the individual by randomly re-assigning each dimension data of the individual within the range of the upper and lower limits corresponding to the dimension data. After the current individual is updated, instead of calculating its objective function value through Equation (15), directly set its objective function value to 10 10 , if a current individual can satisfy all the constraint conditions at the same time, the individual remains unchanged. At this time, the 0th generation population is obtained. Denote the individual with the smallest objective function value among the individuals that can satisfy all the constraint conditions in the 0th generation population as gBest, and denote the objective function value of the individual gBest as F(gBest). Set the globally optimal individual of the ii-th individual and denote it as pBest ii , and adopt X in the 0th generation population ii to initialize pBest ii . Denote the objective function value of pBest ii as F(pBest ii ); S2.3, set the iteration number variable t and the maximum iteration number T, initialize the iteration number t to 1, and set the maximum iteration number T to 1000; S2.4, perform the t-th iteration on the population to obtain the t-th generation population, and the specific process is as follows: S2.4.1, set the individual number i and initialize the individual number i to 1; S2.4.

2. Update the i-th individual to obtain the i-th individual X(t) of the t-th generation population. The specific process is as follows: i (t), and the specific process is as follows: S2.4.2.

1. Set the energy factor of the $i$-th individual at the $t$-th iteration as Calculate the energy factor of the $i$-th individual at the $t$-th iteration according to Equation (17) This parameter is used for the switching of the algorithm search mode: Among them, represents the random number of the \(i\)-th individual at the \(t\)-th iteration, which is generated by a random function, and S2.4.2.

2. If then go to step S2.4.2.3; if then first set the value of X i (t) to be X i (t - 1), and then go to step S2.4.2.

4. X i (t - 1) is the i-th individual of the (t - 1)-th generation population, and || is the absolute value symbol; S2.4.2.

3. Generate a random number using a random number function and If and Update the i-th individual according to the formula shown in Equation (18) to obtain X i (t); If and Update the i-th individual according to the formula shown in Equation (23) to obtain X i (t); If and Update the i-th individual according to the formula shown in Equation (24) to obtain X i (t); If and Update the i-th individual according to the formula shown in Equation (28) to obtain X i (t); After obtaining X i (t) using Equation (18), (23), (24), or (28), calculate the objective function value of X i (t) according to Equation (15). If the objective function value of X i (t) is less than or equal to the objective function value of X i (t - 1), then keep X i (t) unchanged. If the objective function value of X i (t) is greater than the objective function value of X i (t - 1), then update X i (t) to be equal to X i (t - 1); Substitute the data of X i (t) into Equations (2) to (8) to determine whether X i (t) simultaneously satisfies the constraint conditions of Equations (2) to (8). If it does, then keep X i (t) unchanged. Otherwise, update X i (t) again, and re-randomly assign each dimension data of X i (t) within the range of the upper and lower limits corresponding to that dimension data. After the current update of X i (t), instead of calculating its objective function value F(X i (t)) using Equation (15), directly set its objective function value F(X i (t)) to 10 10 ; Among them, X i (t) represents the i-th individual in the t-th generation population, represents the four intermediate individuals generated for the i-th individual at the t-th iteration, || represents the absolute value symbol, and the value of θ is 1.

5. It represents the first 1-row D-dimensional random number vector generated for the ith individual in the tth iteration. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. It represents the second 1-row D-dimensional random number vector generated for the ith individual in the tth iteration. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. It represents the third 1-row D-dimensional random number vector generated for the ith individual in the tth iteration. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. It represents the random number generated for the i-th individual in the t-th iteration, which is used to perturb the current optimal individual to improve the diversity of the population. LF() is the Levy flight function. Indicates the t-th iteration to solve the i-th individual X 1 (t-1),X 2 (t-1),…,X N The average value of (t-1) is given by equation (26), where Γ is the gamma function, Indicates a D-dimensional random number vector with one row generated for the i-th individual in the t-th iteration. The lower limit of each dimension of the random number vector is 0 and the upper limit is 1. If X i The objective function value F(X i (t)) is set equal to 10 10 , then directly obtain X i The objective function value F(X i (t)), if X i The objective function value F(X i (t)) is not set equal to 10 10 , then according to formula (15) calculate its objective function value F(X i (t)), and then make the following judgment: If F(X i (t)) <F(gBest),则使用X i (t) Update the variable value of gBest, otherwise do not update the variable value of gBest; if F(X i (t)) <F(pBest i ), then use X i (t) Update pBest i Otherwise, pBest is not updated. i Variable value; so far, X i (t) Update completed, go to step S2.4.2.4; S2.4.2.

4. Set four intermediate individuals and Calculate two intermediate individuals according to equations (29) and (30) and Among them, is a 1-row D-dimensional random number vector generated by a random function. The lower limit of each dimension data of this random number vector is 0, and the upper limit is 1. is a 1-row D-dimensional random number vector generated by a random function. The lower limit of each dimension data of this random number vector is 0, and the upper limit is 1. ~ is a bitwise negation operator. are any four distinct integers within the range of [1, N] and excluding i. Substitute and into equations (31) and (32) respectively, to obtain two intermediate individuals and Among them, is a 1-row D-dimensional random number vector generated by a random function. The lower limit of each dimension data of this random number vector is 0, and the upper limit is 1. is a 1-row D-dimensional random number vector generated by a random function. The lower limit of each dimension data of this random number vector is 0, and the upper limit is 1. is a random number generated by a random function, and is a random number generated by a random function, and e is the Napier constant, with a value of 2.718281828459045. Substitute the data of and into equations (2) to (8) respectively, and determine whether and simultaneously satisfy the constraint conditions of equations (2) to (8). If and do not simultaneously satisfy, update the intermediate individual that does not satisfy simultaneously once. Randomly re - assign each dimension data of it within the range of the upper and lower limits corresponding to this dimension data, and directly set its objective function value to 10 10 , then substitute it into equation (33) to obtain the intermediate individual If and both satisfy, then directly substitute and into equation (33) to obtain the intermediate individual S2.4.2.5, Use Update X i (t), calculate X using Equation (15) i (t)'s objective function value. If the objective function value of X i (t) is less than the objective function value of gBest, then use X i (t) to update the value of gBest, otherwise do not update gBest. If the objective function value of X i (t) is less than i the objective function value of pBest i (t) to update pBest i , otherwise do not update pBest i ; S2.4.

3. Determine whether the current value of i is equal to N. If not, update the value of i with the sum of the current value of i plus 1, and then return to step S2.4.2 to update the next individual. If it is equal to N, the t-th iteration is completed, and N individuals X 1 (t) to X N (t) are obtained. At this time, proceed to the next step; S2.4.4, determine whether the current value of t is equal to T. If it is not equal, update the value of t with the sum of the current value of t plus 1, and then return to step S2.4 for the next iteration. If it is equal to T, enter the next step; S2.5, output the currently obtained gBest, and the currently obtained gBest is the key parameter for optimizing the weight of the disc spring obtained by the solution.

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