A Particle Collaborative Optimization Method for Pressure Vessel Design
By decomposing the pressure vessel design problem into system-level and discipline-level problems, and introducing improved slack variables and dynamic coefficients, and optimizing using particle swarm update rules, the standard collaborative optimization algorithm is solved, and the global optimization capability and convergence speed are improved.
Patent Information
- Application Number
- CN202111581627.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2021-12-22
- Publication Date
- 2025-06-17
- Estimated Expiration
- 2041-12-22
AI Technical Summary
The standard collaborative optimization algorithm is sensitive to the selection of initial values, with large calculation volume and slow convergence speed, resulting in poor global optimization capabilities.
By decomposing pressure vessel design problems into system-level and discipline-level problems, we introduce improved slack variables and dynamic coefficients, and optimize using particle swarm update rules.
It improves global optimization capability and convergence speed, reduces sensitivity to initial values, and reduces calculation amount and convergence time.
Smart Images
Figure CN114528745B_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of application of optimization algorithms, and specifically relates to a particle collaborative optimization method for pressure vessel design. Technical Background
[0002] Multidisciplinary design optimization (MDO) was proposed by Sobieski, which is a design method based on the mutual cooperation mechanism within the system. MDO can generally be divided into two categories: single-level optimization algorithms and multi-level optimization algorithms. Among these methods, collaborative optimization (CO) has been widely applied in engineering design because it can maintain the autonomy of each disciplinary analysis and the parallelism of processing. When solving problems, CO generally divides the problem into a system-level problem and several disciplinary-level problems for solution. The system-level coordinates the optimization results of each disciplinary-level through its own consistency equality constraints, while the optimal values of the disciplinary-level need to match the shared variables from the system-level. At the same time, the associations between disciplinary-levels also affect the calculation results. Under multiple iterations of this principle, the global optimal value is obtained. However, CO itself still has deficiencies, such as high sensitivity to initial values, large computational volume, and easy to fall into local optimal regions, etc. Summary of the Invention
[0003] The purpose of the present invention is to provide a particle collaborative optimization method for pressure vessel design, which can solve problems such as the sensitivity of the standard collaborative optimization algorithm to the selection of initial values, large computational volume, and slow convergence speed, and improve the global optimization ability and convergence speed.
[0004] The technical solution adopted by the present invention to solve the above problems is a particle collaborative optimization method for pressure vessel design, which decomposes the design problem of the pressure vessel into a system-level problem and several disciplinary-level problems. The system-level problem is the total cost of the pressure vessel, and the disciplinary-level problems are established according to the relationship between constraint conditions. The optimization steps are as follows:
[0005] S1: According to the system-level and disciplinary-level models of pressure vessel design.
[0006] S2: Initialize the relevant parameters of the system-level and disciplinary-level. These parameters include the initial variables z0 and x of the system-level and disciplinary-level i0 , the static relaxation part dv of the system-level improved relaxation variable ε, and the constant w in the disciplinary-level weight coefficient α.
[0007] S3: Set the relevant parameters of the particle swarm update rule, including the initial velocity v, learning factors c1 and c2, random numbers rand1 and rand2, the upper and lower limits v max and v min of the particle velocity, the comprehensive inconsistency information Δ c and the system-level inconsistency information Δ s。
[0008] S4: The system level passes the optimal solution to the discipline level (the initial optimal solution is the initial variable z0), adds the dynamic coefficient α*F i (x i ) to the discipline-level objective function, and obtains the discipline-level optimal solution and optimal value under the condition of satisfying its own constraints. This optimal solution will be used as prior knowledge to become the initial solution for the next discipline-level startup.
[0009] S5: Each discipline level passes the optimal solution to the system level, adds the improved slack variable ε to the system-level constraint conditions, calculates the optimal solution and optimal value, and updates the optimal solution using the particle swarm update rule to obtain the initial solution for the next system-level startup.
[0010] S6: Calculate the global convergence condition Observe whether it is satisfied δ is an extremely small real number, and the value range of δ is 10 -7 ~10 -3 , F k (z) is the objective function value at the kth time. If the condition is satisfied, jump out of the loop to obtain the global optimal solution; otherwise, execute steps 4 and 5 and continue to update the relevant parameters.
[0011] As an optimized version of the above solution, the system-level model formula with the improved slack variable is as follows:
[0012]
[0013] Among them, F(z) is the objective function of the system-level problem, that is, the cost problem of the pressure vessel, z is the system-level variable, that is, the global variable; J i (z) is the consistency equality constraint, z j is the jth global variable, s i is the number of global variables under the ith discipline level, is the optimal value of the jth shared variable from the ith discipline-level problem.
[0014] As an optimized version of the above solution, the formula for the improved slack variable ε is as follows:
[0015] ε = (λ k *Δ c ) 2 +dv
[0016]
[0017] Δ c = max(Δ1, Δ2)
[0018] Δ1 = max(||x i -xi′ ||), i ≠ i'
[0019] Δ2 = max(||z k - x i ||)
[0020] where (λ k * Δ c ) 2 is the dynamic relaxation part, and λ k is a dynamically decreasing constant. dv is the static relaxation part. Δ c is the comprehensive inconsistency information, which takes the maximum value of Δ1 and Δ2. Δ1 is the maximum inconsistency information between disciplines, and Δ2 is the maximum inconsistency information between the system level and the discipline level.
[0021] As an optimized solution to the above scheme, the formula for the particle swarm update rule to update the system-level optimal solution is as follows:
[0022] v k+1 = v k + c1 * rand1 * (pbest k - z k ) + c2 * rand2 * (gbest k - z k )
[0023] Δ s = ||z k - z k-1 ||
[0024] z k+1 = z k + v k+1
[0025] v min ≤ v k ≤ v max
[0026]
[0027] where v k and v k+1 are the current velocity and the next velocity of the particle, respectively, and k is the number of iterations. c1 and c2 are learning factors, and rand1 and rand2 are random numbers between 0 and 1. The self-cognition term (pbest k - z k ) is determined by the comprehensive inconsistency information Δ C , and pbest k is the optimal solution of the current particle; the swarm-cognition term (gbest k - z k ) is determined by the system-level inconsistency information Δs Determine that (gbest k -z k ) is the global optimal solution of the particle. v max and v min are the upper and lower limits of the particle velocity and can dynamically narrow the range.
[0028] v k+1 = v k + c1 * rand1 * (pbest k -z k ) + c2 * rand2 * (gbest k -z k ) can be simplified to: v k+1 = v k + c1 * rand1 * Δ c + c2 * rand2 * Δ s .
[0029] As an optimized solution of the above scheme, the formula of the discipline-level model with dynamic coefficients is as follows:
[0030] min J′ i (x i ) = J i (x i ) + α * F i (x i )
[0031] s.t. g i (x i ) ≤ 0
[0032]
[0033] Among them, J i (x i ) is the objective function of the original i-th discipline level, g i (x i ) is the constraint function of the discipline-level problem, J′ i (x i ) is the improved objective function of the i-th discipline level, α * F i (x i ) is the discipline-level dynamic coefficient, and the formula is as follows:
[0034]
[0035] 10 n-1 ≤ w ≤ 10 n
[0036] α is the weight coefficient, which is determined by the comprehensive inconsistency information Δ c and the system-level inconsistency information Δ sJoint decision, take Δ c and Δ s Take the minimum value. w is a fixed constant, and n is the order of magnitude of the estimated target result. F i (x i ) is the system-level objective function value based on the discipline-level optimal value.
[0037] Preferably, the total cost of the pressure vessel includes material, shaping, and welding costs.
[0038] The beneficial effects of the present invention are as follows: Compared with standard collaborative optimization, in the problems of poor sensitivity to initial value selection, poor global convergence ability, and slow convergence speed of the present invention, at the system level, by introducing an improved relaxation variable, the target can reach the global optimal region faster and better, and the search range can be expanded in the global optimal region through the particle swarm update rule. By introducing a dynamic coefficient into the discipline-level objective function, the convergence speed in the early stage and the global convergence ability in the later stage of the discipline level are improved. Using prior knowledge for update can enable the discipline level to start quickly and reduce the convergence time. Description of the Drawings
[0039] Figure 1 It is a schematic flowchart of a particle collaborative optimization method for a pressure vessel design in an embodiment. Detailed Embodiment
[0040] The technical solution of the present invention will be further described below through embodiments and in conjunction with the drawings.
[0041] Embodiment:
[0042] An embodiment of a particle collaborative optimization method for a pressure vessel design, as Figure 1 shown, includes the following steps: S1: Decompose and design the original mathematical model into a system-level and several discipline-level models according to the relevant parameters and some requirements of the pressure vessel.
[0043] S2: Set the parameters that need to be initialized for the system level, discipline level, and improvement part. For example, the system level needs to set the initial variable z0 and the static relaxation part dv of the improved relaxation variable ε in the system-level constraint, and the discipline level needs to set the initial variable x i0 and the fixed constant w added to the weight coefficient α in the objective function.
[0044] S3: Set the relevant parameters of the system-level particle swarm update rule, including the initial velocity u, learning factors c1 and c2, random numbers rand1 and rand2, the upper and lower limits v max and v min of the particle velocity, the comprehensive inconsistency information Δ c and the system-level inconsistency information Δ s .
[0045] S4: The system level passes the optimal solution to the discipline level for solution. The initial optimal solution at the system level can be set as the initial variable z0, which will be used as the target solution when the discipline level performs the solution, and a dynamic coefficient α*F is added. i (x i ), and the dynamic coefficient is composed of the weight coefficient (10 n-1 ≤w≤10 n , where n is the order of magnitude of the estimated target result) and the objective function value F i (x i ) calculated based on the optimal solution at the discipline level. Its role is to dynamically change the convergence target in the early stage of convergence, making it easier to achieve the convergence effect. In the later stage of convergence, as the dynamic coefficient decreases, the global optimization ability will also be enhanced. After satisfying the discipline-level constraint conditions, the optimal solution at the discipline level is obtained, and this solution will become the initial solution for the next discipline-level startup in the form of prior knowledge, so as to achieve the effect of rapid startup at the discipline level.
[0046] The formula for the discipline-level model with the added dynamic coefficient is:
[0047] minJ′ i (x i ) = J i (x i ) + α*F i (x i ), i = 1, 2
[0048] s.t.g i (x i ) ≤ 0
[0049]
[0050] Among them, J i (x i ) represents the objective function of the original i-th discipline level, g i (x i ) is the constraint function of the discipline-level problem, and J′ i (x i ) is the improved objective function of the i-th discipline level. In this embodiment, s.t.g i (x i ) ≤ 0 is only for example. In actual applications, the expression can be selected according to the actual situation and can be expressed as:
[0051] s.t.g 11 (x) = x 11 + 0.0193x 13 ≤ 0
[0052] g 12 (x) = x 12+0.0954x 13 ≤0
[0053] 1≤x 11 ≤99
[0054] 1≤x 12 ≤99
[0055] 10≤x 13 ≤200
[0056]
[0057] g 24 (x)=x 24 -240≤0
[0058] 10≤x 23 ≤200
[0059] 10≤x 24 ≤200。
[0060] S5: Each discipline level passes the optimal solution to the system level, adds the improved slack variable ε to the system-level constraint conditions, calculates the optimal solution and the optimal value, and uses the update rule of the particle swarm to update the optimal solution. The formula for the particle swarm update rule is:
[0061]
[0062] where, v k and v k+1 are the current speed and the next speed of the particle, k is the number of iterations. c1 and c2 are learning factors, and rand1 and rand2 are random numbers between 0 and 1. The self-cognition term (pbest k -z k ) is determined by the comprehensive inconsistency information Δ C , and pbest k is the optimal solution of the current particle; the swarm cognition term (gbest k -z k ) is determined by the system-level inconsistency information Δ s , and (gbest k -z k ) is the global optimal solution of the particle. v max and v min are the upper and lower limits of the particle speed, which can dynamically narrow the range. According to the relationship between the self-cognition term, the swarm cognition term and Δ C , Δ s , the formula v k+1 =v k +c1*rand1*(pbest k -z k) + c2 * rand2 * (gbest k - z k ) can be simplified to: v k+1 = v k + c1 * rand1 * Δ c + c2 * rand2 * Δ s .
[0063] The updated optimal solution is used as the initial solution when the next system - level iteration starts. The system - level model formula with the improved slack variable can be expressed as:
[0064]
[0065] Among them, F(z) is the objective function of the system - level problem, and the expression is only for example. In practical applications, the expression can be selected according to the actual situation; the system - level problem is the cost problem of the pressure vessel, z is the system - level variable, that is, the global variable; J i (z) is the consistency equality constraint, z j is the j - th global variable, s i is the number of global variables at the i - th disciplinary level, is the j - th disciplinary - level optimal value of the i - th disciplinary - level problem.
[0066] The formula for the improved slack variable ε can be expressed as:
[0067] ε=(λ k * Δ c ) 2 + dv
[0068] k is the number of iterations
[0069] (λ k * Δ c ) 2 is the dynamic relaxation part, and du is the static relaxation part. Δ c is the comprehensive inconsistency information, taking the maximum value of Δ1 and Δ2. Δ1 is the maximum inconsistency information between the largest disciplinary levels, and Δ2 is the maximum inconsistency information between the largest system - level and disciplinary - level. Δ1 = max(||x i - x i′ ||), i ≠ i′, Δ2 = max(||z k - x i ||).
[0070] The main function of the dynamic relaxation part is to speed up the convergence speed in the early stage of iteration so that the target variable can quickly enter the global optimal region. The function of the static relaxation part is to ensure a certain convergence speed and global optimization ability in the later stage of iteration. After satisfying the conditions of its own constraints, the system-level optimal solution is obtained, and then the optimal solution is updated through the particle swarm update rule to expand the search range.
[0071] S6: Calculate global convergence conditions Observe whether it is satisfied δ is a very small positive real number. In this embodiment, the value range of δ is 10 -7 ~10 -3 If the conditions are met, the loop will be exited to obtain the global optimal solution, otherwise, steps 4 and 5 will be repeated and relevant parameters will continue to be updated.
[0072] The working principle of the present invention is as follows: On the basis of standard collaborative optimization, at the system level, the present invention transforms the system-level consistency equality constraint into a consistency inequality constraint by introducing a slack variable of comprehensive inconsistent information, thereby accelerating the convergence speed of the early stage of the system-level iteration and the global optimization capability of the later stage of the iteration, and updates the system-level iteration value through the particle swarm update rule, expands the system-level target optimization range, and reduces the probability of the target falling into the local optimum. At the discipline level, this method accelerates the convergence speed of the early stage of the discipline-level iteration by adding a dynamic coefficient to the discipline-level objective function, and accelerates the discipline-level solution speed by using discipline-level prior knowledge. The present invention can effectively solve the problems of sensitivity to the selection of initial values, large amount of calculation and slow convergence speed, and improves the accuracy.
[0073] The specific embodiments described herein are only for further explanation of the present invention. Those skilled in the art may modify, supplement or replace the specific contents described in a similar manner without departing from the spirit of the present invention or exceeding the scope defined by the appended claims.
Claims
1. A particle collaborative optimization method for pressure vessel design, which decomposes the pressure vessel design problem into a system-level problem and several discipline-level problems. The system-level problem is the total cost of the pressure vessel, and the discipline-level problems are established according to the relationship between the constraints. The optimization steps are as follows: S1: Establish a system-level model and a discipline-level model; S2: Initialize the system-level related parameters and the discipline-level related parameters. The system-level related parameters include the initial variable z0 and the static relaxation part dv of the improved relaxation variable ε, and the discipline-level related parameters include the initial variable x i0 and the fixed constant w in the discipline-level weight coefficient α; The improved relaxation variable is the sum of the dynamic relaxation part (λ k *Δ e ) 2 and the static relaxation part dv, where λ k is a dynamically decreasing constant, and k is the number of iterations; Δ c is the comprehensive inconsistency information, which is the larger value taken from "the maximum inconsistency information between disciplines" and "the maximum inconsistency information between the system level and the discipline level"; S3: Set the relevant parameters of the particle swarm update rule. The relevant parameters of the particle swarm update rule include the initial velocity v, learning factors c1 and c2, random numbers rand1 and rand2, and the upper and lower limits v of the particle velocity max and v min , the comprehensive inconsistency information Δ c and the system-level inconsistency information Δ s ; S4: The system level passes the optimal solution to the discipline level, and adds a dynamic coefficient α*F to the discipline-level objective function i (x i ), and takes the sum of the discipline-level objective function and the dynamic coefficient as the improved discipline-level objective function, where α is the weight coefficient, and F i (x i ) is the system-level objective function value based on the discipline-level optimal value, and x i is the i-th discipline-level variable; the discipline-level optimal solution and optimal value are obtained under the condition of satisfying its own constraints, and the discipline-level optimal solution becomes the initial solution for the next discipline-level startup as prior knowledge; S5: Each discipline level passes the optimal solution to the system level, adds an improved slack variable ε to the system-level constraint conditions, calculates the optimal solution and the optimal value, and updates the optimal solution using the particle swarm update rule to obtain the initial solution for the next system-level startup; After adding the improved slack variable ε, the system-level model includes: the minimum value of the objective function of the system-level problem; the sum of squares of the differences between the global variables and the discipline-level optimal values is less than or equal to the improved slack variable ε; S6: Calculate whether the global convergence condition is satisfied. If so, output the global optimal solution; otherwise, execute step S4.
2. The particle collaborative optimization method for pressure vessel design according to claim 1, characterized in that: The formula of the system-level model with the improved slack variable ε in step S5 is as follows: Among them, F(z) is the objective function of the system-level problem, and z is the system-level variable; J i (z) is the consistency equality constraint, and z j is the j-th global variable, and s i is the number of system-level variables at the i-th disciplinary level, is the j-th disciplinary-level optimal value from the i-th disciplinary-level problem.
3. The particle collaborative optimization method for pressure vessel design according to claim 2, characterized in that: The ε is the improved slack variable, and the formula is as follows: Among them, (λ k *Δ c ) 2 is the dynamic relaxation part, λ k is a dynamically decreasing constant, k is the number of iterations; dv is the static relaxation part, Δ c is the comprehensive inconsistency information, Δ1 is the maximum inconsistency information between disciplines, and Δ2 is the maximum inconsistency information between the system level and the discipline level.
4. The particle collaborative optimization method for pressure vessel design according to claim 1, characterized in that: The formula of the particle swarm update rule for updating the system-level optimal solution in step S5 is as follows: where v k and v k+1 are the current velocity and the next velocity of the particle, k is the number of iterations, c1 and c2 are learning factors, rand1 and rand2 are random numbers between 0 and 1, z k is the current system-level variable, and z k+1 is the system-level variable applied to the next iteration. The self-cognition term, i.e., pbest k -z k is determined by the comprehensive inconsistency information Δ C , and pbest k is the optimal solution of the current particle; the swarm-cognition term, i.e., gbest k -z k is determined by the system-level inconsistency information Δ s , and gbest k -z k is the global optimal solution of the particle, and v max and v min are the upper and lower limits of the particle velocity.
5. The particle collaborative optimization method for pressure vessel design according to claim 1, characterized in that: The formula of the discipline-level model with dynamic coefficients in step S4 is: Among them, J i (x i ) is the objective function of the original $i$-th disciplinary level, $g i (x i ) is the constraint function of the disciplinary-level problem, $J' i (x i ) is the improved objective function of the $i$-th disciplinary level, $\alpha * F i (x i ) is the disciplinary-level dynamic coefficient, and the formula is as follows: Among them, α is the weight coefficient, w is a fixed constant, and n is the order of magnitude of the estimated target result.
6. A particle collaborative optimization method for the design of a pressure vessel according to claim 1, characterized in that: The total cost of the pressure vessel includes material, shaping, and welding costs.
7. A particle collaborative optimization method for the design of a pressure vessel according to claim 1, characterized in that: The satisfaction of the global convergence condition described in step S6 means that: wherein, the value range of δ is 10 -7 ~10 -3 , F k (z) is the objective function value at the k-th time.