A discipline-based self-organizing collaborative optimization method for gear transmission systems

Abstract

This invention proposes a discipline-based self-organizing collaborative optimization method for gear transmission systems. The specific steps are as follows: 1. Initialize the expected value z0 of the gear transmission system design variables; 2. Reconstruct the discipline-level optimization mathematical model based on the expected value of the system-level allocated design vectors and the linear approximation of the key constraints from the previous round of other disciplines; 3. The bearing sub-discipline and the gear subsystem are optimized using optimization algorithms according to the constructed discipline-level optimization model, obtaining their respective optimization solutions and returning them to the system level, while simultaneously updating their respective discipline state variables y. i ; 4: Compare the expected value of the design vector with the disciplinary solution and the system-level allocation to perform system-level optimization and update the current round 5: Convergence verification; if the convergence condition is met, terminate the optimization; otherwise, go to 2 to continue the next round of optimization; 6: End, output the optimal objective function value and the values ​​of each design variable when the objective function value is obtained.

Description

Technical Field

[0001] This invention provides a discipline-based self-organizing collaborative optimization method for gear transmission systems, belonging to the field of multidisciplinary design and optimization. Background Technology

[0002] With the continuous development and improvement of science and technology and human needs, the complexity of engineering systems is increasing. The design and development of complex engineering systems involves multiple coupled disciplines, resulting in computational complexity. Multidisciplinary Design Optimization (MDO), which emerged in the 1980s, is an effective method for solving optimization problems in complex engineering systems. It strives to balance the various disciplines at each stage of the design process, fully considers the mutual influence and coupling between disciplines, adopts effective optimization strategies and distributed computing technology, flexibly organizes and manages the entire system design process, and aims to obtain the overall optimal solution of the system by fully utilizing the synergistic effects generated by the interaction between disciplines. This achieves the goals of improving the performance of complex engineering systems, reducing design costs, and shortening the development cycle.

[0003] Multidisciplinary design optimization strategies are the core research content of MDO and one of the most active research areas. Among them, collaborative optimization strategies are widely used in aerospace, shipbuilding, automotive, and mechanical fields due to their unique characteristics such as high degree of disciplinary autonomy, multi-level optimization, and distributed computing. At the same time, collaborative optimization strategies also suffer from low computational efficiency and convergence difficulties due to their inherent mathematical model definition and optimization mechanism. To this end, this invention proposes a discipline self-organizing collaborative optimization strategy, which modifies the basic structure of collaborative optimization strategies so that the optimization of subsystems can act on the global goal under the conditions of satisfying the constraints of the discipline and the consistency constraints between disciplines. Its main contents are two points: (1) A virtual system-level coordinated optimization model is constructed. The system-level design point of the new round is updated by the relative distance between the previous system-level optimization point and the subsystem design point, replacing the traditional expensive system-level optimization, thereby improving the efficiency of system-level coordinated optimization; by establishing a new information channel between two subsystems to transmit state variables, one subsystem can be optimized as independently as possible with the fuzzy help of the other subsystem, thereby ensuring the high degree of disciplinary autonomy of the subsystem and improving the computational efficiency and convergence performance of handling large-scale MDO problems. (2) For the MDO problem with dispersed constraint space between disciplines, a discipline self-organizing collaborative optimization strategy based on local linear approximation subspace is proposed. By constructing a linear approximation of the key constraints of disciplines to restrict the constraint space of other disciplines, the influence of constraints between disciplines is controlled, thereby improving the convergence speed of collaborative optimization.

[0004] Collaborative optimization strategies are highly applicable to engineering and easy to organize, manage, and implement; however, their shortcomings in optimization speed and convergence effectiveness hinder their application in engineering design. The method proposed in this invention addresses these theoretical deficiencies by conducting in-depth research, further enriching and improving collaborative optimization strategies. This is of great significance for promoting the in-depth application of multidisciplinary optimization techniques in complex product design processes, shortening development cycles, and enhancing product competitiveness. Summary of the Invention

[0005] (I) Purpose of the present invention The purpose of this invention is to provide a discipline-based self-organizing collaborative optimization method. This method fundamentally addresses the shortcomings of existing collaborative optimization methods by modifying the basic structure of standard collaborative optimization methods and reducing the variable space dimensionality of system-level optimization problems. The overall process of the method is shown in the attached figure. Figure 1 As shown.

[0006] (II) Technical Solution This invention introduces a discipline-based self-organizing collaborative optimization method for gear transmission systems, the specific steps of which are as follows: Step 1: Initialize the expected value z0 of the design variables for the gear transmission system.

[0007] Step 2: Based on the expected value of the design vector allocated at the system level. Linear approximation of key constraints in the previous round of other disciplines Reconstruct subject-level optimized mathematical models.

[0008] Step 3: The bearing sub-discipline and gear subsystem are optimized using optimization algorithms based on the constructed discipline-level optimization model to obtain the optimized solutions for their respective disciplines. And return to the system level, while updating the respective subject-specific state variables. .

[0009] Step 4: Comparative Study and the expected value of the design vector at the system level. System-level optimization and updates were performed to obtain the results of this round. .

[0010] Step 5: Convergence Verification. If the convergence condition is met, the optimization is terminated; otherwise, proceed to Step 2 to continue the next round of optimization.

[0011] Step 6: End.

[0012] The technical solution of the present invention will be further described in detail below: In step 1, the expected values ​​of the design variables for the gear transmission system are initialized as follows: z is a system-level design variable vector, set in the first round of optimization. This represents the mean of all design variables.

[0013] In step 2, the subject-level optimization mathematical model reconstruction specifically involves: (1) Linear approximation of key constraint functions The basic idea of ​​linear approximation of key constraint functions is that during the optimization process, the key constraint boundary of the discipline is passed to other disciplines in the form of first-order linear approximation constraints. In this way, during the optimization process at the discipline level, it is ensured that the optimal solution of the discipline is in the solution space formed by the constraints of the discipline, and at the same time, it is guaranteed to the greatest extent that the optimal solution of the discipline does not violate the constraints of other disciplines.

[0014] For collaborative design optimization strategies, design variables X optimal solution It is the point of tangency between the contour line of the discipline consistency constraint function and the critical constraint. (See attached image.) Figure 2 As shown, at the point where the optimal solution is located, the consistency constraint circle and the constraint boundary have the same common tangent. t tangent It indicates. In At this point, the partial derivative of the consistency constraint function with respect to the design variables is equal to the partial derivative of the critical constraint function with respect to the design variables, that is... (1) In the formula, Represents the consistency constraint function. This represents the key constraint function.

[0015] In multidisciplinary optimization processes, solving for the partial derivatives of constraint functions with respect to X requires methods such as finite differences, which are computationally complex. However, consistency constraint functions... The partial derivative at the point is relatively easy to find, for the consistency constraint function at... The partial derivative is obtained as shown in equation (2).

[0016] (2) In the formula, Z represents the expectation of the design variable vector. It can be seen that for discipline-level optimization, only the optimal value of the optimization of this discipline needs to be found to obtain the partial derivative of the consistency constraint function with respect to the design variables at the optimal point. That is, according to formula (3), the tangent direction vector of the key constraint function at the optimal point can be obtained. The tangent of the key constraint function at the optimal point, that is, the first-order linear approximation function of the key constraint function at the optimal point, can be expressed as: (3) Since the first-order linear approximation of the critical constraint function at the optimum is its tangent line at the optimum, it must pass through the optimum. The constant C can be obtained by substituting the coordinates of the optimal point. Thus, the first-order linear approximation function of the critical constraint function at the optimal point can be obtained.

[0017] (2) Definition of Discipline Optimization Model The discipline-level optimization of the discipline-self-organizing collaborative optimization strategy can be expressed as: (4) In the formula, F It is a global optimization goal; For the first i The objective function of each subsystem; It is the first i Shared design variables for optimization across disciplines; It is the first i Local design variables for each discipline; These are state variables, provided by other subsystems; It is a compatibility penalty factor; It is the first i Inequality constraints of each subsystem; This represents the key first-order linear approximation constraints passed from other disciplines to this discipline; It is the first i Equality constraints of each subsystem; These are design variables provided at the virtual system level; , They represent The lower and upper limits; , They represent The lower and upper limits.

[0018] In step 3, the subject-level optimization calculation specifically involves: Step 1: Randomly generate the initial population, calculate the fitness of individuals (using a subject-specific objective function as fitness), and find the current best individual.

[0019] Step 2: If the current best individual meets the convergence condition, terminate the optimization; otherwise, continue to step 3.

[0020] Step 3: Set the current iteration number gen=1.

[0021] Step 4: If gen < maxgen (maxgen is the maximum number of steps to obtain the genetic algorithm), proceed to step 5; Step 5: Based on the roulette wheel selection method, select individuals that meet the selection ratio.

[0022] Step 6 Update the adaptive crossover probability according to formula (5) P c New individuals are generated through cross-fertilization.

[0023] (5) In the formula, f max The maximum fitness value in the current population. f avg It is the average fitness value of the current population. P cini It is the initial crossover probability. P cmax and P cmin These represent the upper and lower limits of the crossover probability, respectively. f big This indicates the higher fitness among the two individuals to be crossed. I c This indicates the relational information needed to adjust the crossover probability.

[0024] Step 7 Update the adaptive mutation probability according to formula (6) P m The mutation generates a new individual.

[0025] (6) In the formula, f max The maximum fitness value in the current population. f avg It is the average fitness value of the current population. P mini It is the initial mutation probability. P mmax and P mmin These represent the upper and lower limits of the mutation probability, respectively. f w Indicates the fitness of the individual to be mutated. I m This represents the relational information needed to adjust the mutation probability.

[0026] Step 8: Update the population and set gen+1. Proceed to Step 4. Step 9: Output the optimization results.

[0027] In step 4, the system-level optimization specifically involves: System-level optimization model definition: The role of virtual system level is to provide system design variables. To balance the inconsistencies between subsystems, the "virtual" approach, as opposed to the standard CO strategy, uses a coordination model at the system level that is not a true optimization model but rather a harmonization model. To accelerate the convergence of shared variables in the design optimization schemes of each subsystem, this invention proposes a system-level coordination model for solving system design variables using non-optimization search, as shown in the appendix. Figure 3As shown.

[0028] Figure 3 Taking two-dimensional space as an example, the principle can be extended to three-dimensional or higher-dimensional spaces. Refer to the convergence criteria generally used in optimization. , Represents system-level design variables. If is any small positive number, then the closer to the global optimum, the smaller the movement distance. Based on the geometric positional relationship between the system-level design point and the subsystem-level optimization point, if Compare The smaller value indicates that the optimization point of subsystem 1 in this round is closer to the global optimum, at which point the system-level design point is taken. and The midpoint of the coordinates is clearly too conservative, and subsystem 2 should make more compromises. Therefore, it is more appropriate to use relative distances to coordinate the values ​​of system-level design points. Thus, the virtual system-level model can be defined as follows: (7) In the formula It is the first System design variables for the next cycle; and It is the first Shared design variables obtained from the optimization of subsystem 1 and subsystem 2 in the next iteration.

[0029] In step 5, the convergence verification specifically involves: If the convergence condition is met ( , ε 1 and ε If both 2 are any small positive numbers, then the optimization will terminate; otherwise, proceed to step 2 to continue the next round of optimization. Indicates the first The fitness value in the next cycle. Indicates the first The fitness value in the next cycle. Indicates the first Shared design variables of subsystems in the next loop.

[0030] In step 6, the process ends, and the optimal objective function value and the values ​​of each design variable when the objective function value is obtained are output.

[0031] The advantages and beneficial effects of this invention are as follows: This invention applies to the collaborative optimization of gear transmissions in large and complex equipment such as aerospace, shipbuilding, automotive, and machinery. This method fully considers the problems of low computational efficiency and convergence difficulties inherent in traditional collaborative optimization strategies due to their inherent mathematical model definitions and optimization mechanisms. It proposes a discipline-based self-organizing collaborative optimization strategy, modifying the basic structure of the strategy. This allows the optimization of subsystems to act on the global objective while satisfying both discipline-specific constraints and interdisciplinary consistency constraints. A virtual system-level coordinated optimization model is constructed, whose essence is not optimization but coordination of shared variables among subsystems. Simple coordinate operations are used to balance inconsistencies across multiple disciplines, improving the efficiency of system-level coordinated optimization. Subsystem optimization objectives directly affect the global objective, reducing the number of iterations at both the system and subsystem levels and improving the overall computational efficiency. Multidisciplinary consistency constraints are calculated using distance dependency, and state variables are obtained through information channels between subsystems rather than being provided by the system level as system design variables. This significantly reduces the dimensionality of the design variable space and improves convergence performance. To address the MDO problem with dispersed constraints across disciplines, a self-organizing collaborative optimization strategy based on local linear approximation subspaces is proposed. This strategy involves constructing a linear approximation of the key constraints of another discipline and incorporating it into other disciplines to control the influence of constraints between disciplines, thereby improving the convergence speed of collaborative optimization. Attached Figure Description

[0032] Figure 1 This is the overall flowchart of the method described in this invention.

[0033] Figure 2 This is a schematic diagram of the linear approximation of the key constraint function.

[0034] Figure 3 This is a schematic diagram showing the relative positions of system-level optimization points and subsystem optimization points.

[0035] Figure 4 This is a schematic diagram of the spatial relationship of design variables in an aircraft gear transmission system.

[0036] Figure 5 This is a structural diagram of an aircraft gear transmission system. Detailed Implementation

[0037] The invention will be further explained below using an aviation gear transmission system as an example.

[0038] This invention considers a self-organizing collaborative optimization approach based on linear approximation of key constraint functions to optimize the design of aerospace gear transmission systems. As a common mechanism installed between the propeller and piston engine of small aircraft, the aerospace gear transmission system transmits rotation between the two, outputting a suitable rotational speed to obtain maximum output power. The goal of multidisciplinary design optimization of aerospace gear transmission systems is to achieve the minimum volume (lightest weight for a given material density) of the aerospace gear transmission system while satisfying numerous constraints on gears and shafts in the transmission mechanism. This optimization problem includes seven design variables: The width of the tooth surface. For gear module, This refers to the number of teeth on the pinion. and For bearing spacing, and The spatial relationship between the variables is as follows: (This refers to the distance between the large and small gear shafts.) Figure 4 As shown.

[0039] The objective function of the optimization problem is shown in Equation (8).

[0040] (8) The constraints to be considered during function optimization are shown in equations (9)-(19). Table 2 shows the maximum bending stress of the gear. The maximum contact stress of the gears must not exceed the specified value. The design values ​​must be met; and This refers to the maximum lateral deflection of the large and small gear shafts, which must not exceed the specified value. and This refers to the maximum internal stress on the shafts of the large and small gears, which must meet strength requirements; the gear size must also meet... , and Size and space constraints; and It is an empirical formula for calculating gear shaft dimensions.

[0041] (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) In addition, each variable is subject to upper and lower bound constraints. The design variables and their upper and lower bounds for the aerospace gear transmission system are shown in Table 1.

[0042] Table 1 Design Variables for Aircraft Gear Transmission Systems This optimization problem includes two subsystems: gears and bearings, such as... Figure 5 As shown. Design variables , , and constraints , , , , Belongs to the gear subsystem, design variables , , , and constraints , , , , , It belongs to the bearing subsystem.

[0043] Step 1: According to the steps in the technical solution, four initial points are selected as the expected values ​​of the design variables to study the optimization problem of the gear reducer. The initial points are shown in Table 2.

[0044] Table 2 Initial point settings for gear transmission system Step 2: Using the expected values ​​of the design variables from Step 1 and the provided constraints for each discipline, extract the key constraints at the discipline level and treat them as supplementary constraints for other disciplines. Adopt the idea of ​​linear approximation of key constraints to redefine the discipline-level optimization model; that is, when optimizing the gear subsystem, add the key constraint function passed from the bearing subsystem to the constraints, which is the key first-order linear approximation constraint; the same applies when optimizing the bearing subsystem.

[0045] Step 3: Use an adaptive genetic algorithm to optimize the gear subsystem and bearing subsystem respectively, and return the optimization results to the system level.

[0046] Step 4: Compare the solutions of each discipline in this round with the expected values ​​of the design variables of the system-level allocation in the previous round, and update them using formula (7) to obtain the expected values ​​of the system-level design variables in this round.

[0047] Step 5: Determine if the optimization process has converged, based on the convergence criteria ( , and The algorithm checks if any small positive numbers are met. If so, the optimization process terminates; otherwise, it proceeds to step two for the next round of optimization. (The convergence condition is...) Indicates the first The fitness value in the next cycle. Indicates the first The fitness value in the next cycle. Indicates the first Shared design variables of subsystems in the next loop.

[0048] Step 6: For the four initial points, output the optimization results and the number of iterations, as shown in Table 3.

[0049] Table 3 Optimization Results

[0050] As can be seen from Table 3, the optimization results of the four selected initial mean points all converged to the vicinity of the theoretical optimal solution, obtaining a relatively ideal optimal solution. Moreover, all of them converged to the ideal solution in only 8 to 24 iterations.

[0051] Step 7: Based on the above results, obtain the minimum volume of the aircraft gear transmission system (the lightest mass when the manufacturing material density is constant). This provides a theoretical reference for the design of aircraft gear transmission systems.

Claims

1. A discipline-based self-organizing collaborative optimization method for gear transmission systems, characterized in that: The specific steps are as follows: Step 1: Initialize the expected value z0 of the design variables for the gear transmission system; Step 2: Based on the expected value of the design vector allocated at the system level. Linear approximation of key constraints in the previous round of other disciplines Reconstructing subject-specific optimized mathematical models; Step 3: The bearing sub-discipline and gear subsystem are optimized using optimization algorithms based on the constructed discipline-level optimization model to obtain the optimized solutions for their respective disciplines. And return to the system level, while updating the respective subject-specific state variables. ; Step 4: Comparative Study and the expected value of the design vector at the system level. System-level optimization and updates were performed to obtain the results of this round. ; Step 5: Convergence verification; if the convergence condition is met, terminate the optimization; otherwise, proceed to step 2 to continue the next round of optimization. Step 6: End, output the optimal objective function value and the values ​​of each design variable when the objective function value is obtained; Specifically, in step 2, the subject-level optimization mathematical model reconstruction is as follows: (1) Linear approximation of key constraint functions For collaborative design optimization strategies, design variables X optimal solution This is the point of tangency between the contour line of the subject consistency constraint function and the key constraint; at the point where the optimal solution is located, the consistency constraint circle and the constraint boundary have the same common tangent line. t tangent Indicate; at At this point, the partial derivative of the consistency constraint function with respect to the design variables is equal to the partial derivative of the critical constraint function with respect to the design variables, that is... （1） In the formula, Let g represent the consistency constraint function. c Represents the key constraint function; In multidisciplinary optimization processes, solving the partial derivatives of constraint functions with respect to X requires the use of finite difference methods, which are computationally complex. However, consistency constraint functions... The partial derivative at the point is easy to find, for the consistency constraint function at... The partial derivative is obtained as shown in equation (2); （2） In the formula, Z represents the expectation of the design variable vector. The tangent direction vector of the critical constraint function at the optimal point is obtained according to formula (3). The tangent of the critical constraint function at the optimal point, which is also the first-order linear approximation function of the critical constraint function at the optimal point, is expressed as: （3） Since the first-order linear approximation of the critical constraint function at the optimum is its tangent line at the optimum, it must pass through the optimum. Substituting the coordinates of the optimal point into the equation, we obtain the constant C; thus, we obtain the first-order linear approximation function of the key constraint function at the optimal point. (2) Definition of Discipline Optimization Model The subject-level optimization of the subject-organizing collaborative optimization strategy is represented as follows: （4） In the formula, F It is a global optimization goal; For the first i The objective function of each subsystem; It is the first i Shared design variables for optimization across disciplines; It is the first i Local design variables for each discipline; These are state variables, provided by other subsystems; It is a compatibility penalty factor; It is the first i Inequality constraints of each subsystem; This represents the key first-order linear approximation constraints passed from other disciplines to this discipline; It is the first i Equality constraints of each subsystem; These are design variables provided at the virtual system level; , They represent The lower and upper limits; , They represent The lower and upper limits; In step 4, the system-level optimization specifically involves: System-level optimization model definition: Optimize the convergence criteria , Represents system-level design variables. For any small positive number, the closer to the global optimum, the smaller the movement distance; based on the geometric positional relationship between the system-level design point and the subsystem-level optimization point, if Compare The smaller value indicates that the optimization point of subsystem 1 in this round is closer to the global optimum, at which point the system-level design point is taken. and The midpoint of the coordinates is clearly too conservative, and subsystem 2 should make more compromises; therefore, it is more appropriate to use relative distance to coordinate the values ​​of system-level design points. Thus, the virtual system-level model is defined as follows: (7) In the formula It is the first System design variables for the next cycle; and It is the first The shared design variables obtained from the optimization of subsystem 1 and subsystem 2 in the next iteration.

2. The discipline-based self-organizing collaborative optimization method for gear transmission systems according to claim 1, characterized in that: In step 1, the expected values ​​of the design variables for the gear transmission system are initialized as follows: z is the system-level design variable vector, and the initial optimization settings are as follows: This represents the mean of all design variables.

3. The discipline-based self-organizing collaborative optimization method for gear transmission systems according to claim 1, characterized in that: In step 3, the subject-level optimization calculation specifically involves: Step 1: Randomly generate an initial population, calculate the fitness of individuals, and find the current best individual; Step 2: If the current best individual satisfies the convergence condition, terminate the optimization; otherwise, continue to step 3. Step 3: Set the current iteration number gen=1; Step 4: If gen < maxgen, where maxgen is the maximum number of steps to obtain the genetic algorithm, proceed to step 5; Step 5: Based on the roulette wheel selection method, select individuals that meet the selection ratio; Step 6 Update the adaptive crossover probability according to formula (5) P c New individuals are generated through cross-fertilization; (5) In the formula, f max The maximum fitness value in the current population. f avg It is the average fitness value of the current population. P cini It is the initial crossover probability. P cmax and P cmin These represent the upper and lower limits of the crossover probability, respectively. f big This indicates the higher fitness among the two individuals to be crossed. I c This represents the relational information needed to adjust the crossover probability; Step 7 Update the adaptive mutation probability according to formula (6) P m The mutation generates a new individual; (6) In the formula, f max The maximum fitness value in the current population. f avg It is the average fitness value of the current population. P mini It is the initial mutation probability. P mmax and P mmin These represent the upper and lower limits of the mutation probability, respectively. f w Indicates the fitness of the individual to be mutated. I m This represents the relational information needed to adjust the mutation probability; Step 8: Update the population and set gen+1; Proceed to step 4; Step 9: Output the optimization results.

4. The discipline-based self-organizing collaborative optimization method for gear transmission systems according to claim 1, characterized in that: In step 5, the convergence verification specifically involves: If the convergence condition is met , ε 1 and ε If both 2 are any small positive numbers, then the optimization is terminated; otherwise, proceed to step 2 to continue the next round of optimization. Indicates the first The fitness value in the next cycle. Indicates the first The fitness value in the next cycle. Indicates the first Shared design variables of subsystems in the next loop.

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