Optimization design method of cantilever beam

Abstract

The invention belongs to the technical field of cantilever beam design, and particularly relates to an optimal design method of a cantilever beam, which comprises the following steps of: firstly, establishing a mathematical model aiming at the problem of cantilever beam design; secondly, the algorithm is introduced into an arithmetic optimization algorithm (RAOA) of Q-learning; a random elite pool strategy is introduced to further optimize the algorithm; and finally, carrying out iterative solution through an improved arithmetic optimization algorithm. According to the method, the exploration capability and the convergence speed of the algorithm are enhanced, search strategies with different capabilities are integrated, so that the search effectiveness is improved, a better solution can be obtained in a short time in practical application, and meanwhile, the search strategies can be dynamically adjusted according to different optimization problems to ensure the maximum income.

Description

Technical Field

[0001] The invention belongs to the technical field of cantilever beam design, and in particular relates to an optimization design method for a cantilever beam. Background Art

[0002] Cantilever beam design is a very important aspect in the engineering field. Cantilever beam design involves the analysis and optimization of a beam structure with one end fixed and the other end free. It is commonly found in the fields of construction and bridges. During the design, it is necessary to comprehensively consider the effects of dead load, live load and other loads to ensure that the strength of the beam is sufficient to resist the maximum stress and avoid yield failure; at the same time, the maximum deflection of the beam is controlled to ensure structural rigidity and comfort. In addition, it is necessary to evaluate the overall and local stability of the beam and design reliable connection nodes to ensure the safety and reliability of the structure. The design process must comply with relevant specifications and standards.

[0003] The cantilever beam design algorithm in the prior art has shortcomings such as slow convergence speed and easy to fall into the optimal solution, which reduces the search efficiency of the algorithm. Therefore, an advanced solution method is provided to specifically address the various challenges encountered in the cantilever beam design problem. The core of this method is that it can dynamically adjust the search strategy within the algorithm to adapt to the specific needs of different problems in the optimization process. In this way, whether in the initial exploration stage or the fine optimization stage, the best design solution can be ensured, thereby solving the problem that many existing methods can only be applied to a single scene or a few scenes. Summary of the invention

[0004] The purpose of the present invention is to provide an optimization design method for a cantilever beam, so as to solve the problems existing in the background technology.

[0005] In order to achieve the above technical objectives, the technical solution adopted by the present invention is as follows:

[0006] An optimization design method for a cantilever beam, comprising:

[0007] 1. First, a mathematical model is established for the cantilever beam design problem;

[0008] 2. Secondly, it is introduced into the arithmetic optimization algorithm (RAOA) of Q-learning;

[0009] 3. Then introduce the random elite pool strategy to further optimize the algorithm;

[0010] 4. Finally, the improved arithmetic optimization algorithm is used to iterate and solve the problem;

[0011] The arithmetic optimization algorithm of Q-learning includes: calculating the Q value of each state-action pair through the Q-learning function given in formula (1), and then saving the Q values ​​of all state-action pairs in the Q table:

[0012]

[0013] Among them, for the Q-learning function, it also includes the design of the greedy search parameter ε and the design of the reward function;

[0014] Design of the greedy search parameter ε: ε linearly increases from 0 to 0.9 in the first c (c < T) iterations, and is then fixed at 0.9. Set a random number β4 between 0 and 1. When the random number β4 ≥ ε, perform random search; when the random number β4 < ε, perform greedy search;

[0015] The expression of the reward function is as shown in Equation (2):

[0016]

[0017] Among them, ω is the adjustment factor, t represents the current iteration number, and Best t represents the optimal fitness value of the t-th iteration.

[0018] The steps of the above RAOA algorithm include:

[0019] Step1: Initialize the state space S, action space A, and Q-table;

[0020] Step2: Randomly select an action in the initial state;

[0021] Step3: If β4 > ε, randomly select an action; otherwise, select the action with the maximum reward from the Q-table;

[0022] Step4: Execute the selected action in the AOA algorithm and calculate the reward according to the reward function (2);

[0023] Step5: Update the Q-table using Equation (1);

[0024] Step6: If the termination condition is satisfied, end the loop; otherwise, repeat Steps 3 - 5.

[0025] The Stochastic Elite Pool Strategy Arithmetic Optimization Algorithm includes: exploring the solution space through a variety of different search strategies and integrating search strategies with different capabilities.

[0026] Among them, the various different search strategies in the stochastic elite pool include: The first search strategy:

[0027]

[0028] Where rand is a random number between 0 and 1, X1(c+1) represents the c+1th iteration position obtained using the first search strategy, and X best (c) represents the optimal position of the current iteration, X M (c) represents the average of all solutions in the current population.

[0029] There are many different search strategies in the random elite pool, including: The second search strategy:

[0030] X2(c+1)=(X best (c)-X M (c))×0.1-

[0031] rand+((UB-LB)×rand+LB)×0.1#(5)

[0032] Among them, X2(c+1) represents the c+1th iteration position obtained using the second search strategy, and UB and LB represent the upper and lower bounds of the variables.

[0033] There are many different search strategies in the random elite pool, including: The third search strategy:

[0034] X3(c+1)=(X best (c)-X M (c))-

[0035] rand×((UB-LB)×rand+LB)#(6)

[0036] Among them, X3(c+1) represents the c+1th iteration position obtained using the third search strategy.

[0037] Among them, there are many different search strategies in the random elite pool, including: The fourth search strategy:

[0038] X4(c+1)=X rand (t)-

[0039] rand×|X rand (t)-2×rand×X(t)|#(7)

[0040] Among them, X4(c+1) represents the c+1th iteration position obtained using the fourth search strategy, and X rand (t) represents a random selection of a solution from the elite pool.

[0041] The beneficial effects of the present invention are embodied in the following two aspects:

[0042] (1) The algorithm exploration capability and convergence speed are enhanced, making it more suitable for practical engineering application scenarios. The present invention integrates the REAOA algorithm based on two improved strategies: Q-learning and random elite pool. Not only can the AOA search strategy be dynamically selected according to the changes in the problem, but also the search strategies with different capabilities can be integrated to improve the effectiveness of the search, which is conducive to obtaining better solutions in a short time in practical applications.

[0043] (2) The RAOA algorithm proposed in the present invention can dynamically adjust the search strategy according to different optimization problems to ensure maximum benefits. In view of the shortcomings of arithmetic optimization algorithms such as slow convergence speed and easy to fall into the optimal solution when solving optimization problems, an AOA algorithm (EAOA) based on a random elite pool strategy is proposed. The EAOA algorithm can integrate different search strategies to improve search efficiency. Then, the above two algorithms are integrated to construct an improved AOA algorithm (REAOA) based on the Q-learning random elite pool strategy. A large number of experimental results show that the REAOA algorithm not only inherits the fast convergence characteristics of RAOA in unimodal functions, but also has the optimization ability of the EAOA algorithm in complex multimodal functions, which improves the optimization efficiency and accuracy of REAOA. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The present invention is further illustrated by means of the non-limiting examples given in the accompanying drawings.

[0045] Figure 1 A method step diagram of an optimization design method for a cantilever beam of the present invention;

[0046] Figure 2 A Q-learning algorithm framework diagram of a cantilever beam optimization design method of the present invention;

[0047] Figure 3 A REAOA algorithm flow chart of a cantilever beam optimization design method of the present invention;

[0048] Figure 4 A schematic diagram of a pressure vessel structure of a cantilever beam optimization design method of the present invention;

[0049] Figure 5-31 It is a comparison diagram of the convergence curves of the optimization design method of a cantilever beam of the present invention. DETAILED DESCRIPTION

[0050] In order to enable those skilled in the art to better understand the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0051] like Figure 1-31As shown, the cantilever beam design problem is solved by an improved arithmetic optimization algorithm that integrates Q-learning and random elite pool strategy.

[0052] The main point of the present invention is to add the idea of ​​Q-learning algorithm to arithmetic optimization algorithm (AOA algorithm), construct an AOA algorithm (Reinforcement Arithmetic Optimization Algorithm, RAOA) based on Q-learning, so that AOA can dynamically adjust the search strategy inside the algorithm according to different optimization problems to ensure that the benefits obtained at different optimization stages for different problems are maximized. In addition, it is proposed to integrate the random elite pool strategy into AOA, construct an arithmetic optimization algorithm (Elite Arithmetic Optimization Algorithm, EAOA) based on the random elite pool strategy, integrate search strategies of different capabilities, support each other, and thus improve the effectiveness of the search. In order to make full use of the advantages of high search efficiency of RAOA and high search accuracy of EAOA, so that the advantages of the two complement each other, it is proposed to fuse the two to construct an improved arithmetic optimization algorithm (Reinforcement Elite Arithmetic Optimization Algorithm, REAOA) based on Q-learning and random elite pool strategy

[0053] The improvements made in this method are as follows:

[0054] (1) Q-learning-based arithmetic optimization algorithm (RAOA)

[0055] Design of state space and action space:

[0056] The design of the state space is the basis for the agent to make decisions and evaluate its long-term benefits, and the quality of the state design directly determines whether the Q-learning algorithm can converge. In the intelligent optimization algorithm, we believe that the early stage of the algorithm refers to when the number of iterations is low, and the late stage of the algorithm refers to when the number of iterations is high. In theory, for different optimization problems, the search strategies in the early and late stages of the algorithm should be different. However, when solving different optimization problems, the AOA algorithm is more likely to randomly select search strategies, and it is difficult to guarantee the convergence accuracy and speed of the algorithm. Therefore, the present invention designs a state space based on the number of iterations, that is, each iteration is a state. Utilizing the characteristics of the Q-learning algorithm, for different optimization problems, whether in the early or late stages of the algorithm, each iteration can select the search strategy with the greatest benefit. The action space consists of two actions, namely the multiplication and division search strategy and the addition and subtraction search strategy of the AOA algorithm.

[0057] The table is initially an M×N zero matrix, where M represents the number of states and N represents the number of actions. The Q-values of each state-action pair are calculated by the Q-learning function given in formula (1). Then, the Q-values of all state-action pairs are saved in the Q-table.

[0058]

[0059] Design of the greedy search parameter:

[0060] The present invention designs the greedy search parameter ε as follows. ε linearly increases from 0 to 0.9 in the first c (c < T) iterations and is then fixed at 0.9. A random number β4 between 0 and 1 is set. If β4 ≥ ε, random search is performed. If β4 < ε, greedy search is performed. Then the algorithm is likely to perform random search in the early stage and greedy search in the later stage. This greatly increases the generalization ability and exploration ability of the algorithm.

[0061] Design of the reward function:

[0062] In the Q-learning algorithm, the quality of the algorithm depends to a large extent on the design of the reward function. According to the improvement degree of the optimization result obtained by the algorithm, the agent is given corresponding rewards. The principle of the reward is that the greater the progress, the greater the reward. If there is no change, a penalty will be given. If the fitness value becomes better in the later stage of the algorithm, a greater reward will be given to the agent. In order to make the algorithm more likely to jump out of the local optimum in the later stage, the present invention designs a new reward function, and its mathematical expression is shown in formula (2)

[0063]

[0064] where ω is a regulation factor, t represents the current iteration number, and Best t represents the optimal fitness value at the t-th iteration.

[0065] RAOA algorithm process:

[0066] Step1: Initial state space and action space A and Q-table;

[0067] Step2: Randomly select an action at the initial state;

[0068] Step3: If β4 ≥ ε, randomly select an action, otherwise select the action with the maximum benefit from the Q-table;

[0069] Step4: Execute the selected action in the AOA algorithm and calculate the benefit according to the reward function (2);

[0070] Step 5: Update the Q-table using formula (1);

[0071] Step6: If the termination condition is met, the loop ends, otherwise repeat Step3-Step5.

[0072] (2) Arithmetic Optimization Algorithm (EAOA) based on Random Elite Pool Strategy

[0073] In order to improve the diversity of individuals in the elite pool, the present invention uses four different search strategies to explore the solution space. The purpose is to integrate search strategies of different capabilities and support each other, thereby improving the effectiveness of the search. The capacity of the elite pool is set to Y. During the first Y iterations, the optimal solution of the current iteration is stored in the elite pool each time. If the number of individuals in the elite pool exceeds Y, each elite individual randomly executes a search strategy to further improve the generalization ability of the algorithm. And select the optimal solution in the elite pool to guide the next iteration. Among them, the four search strategies are shown in formulas (3) to (7):

[0074] The first search strategy:

[0075]

[0076] Where rand is a random number between 0 and 1; X1(c+1) represents the c+1th iteration position obtained using the first search strategy; X best (c) represents the optimal position of the current iteration; X M (c) represents the average of all solutions in the current population.

[0077] The second search strategy:

[0078] X2(c+1)=(X best (c)-X M (c))×0.1-rand+((UB-LB)×rand+LB)×0.1#(5)

[0079] Among them, X2(c+1) represents the c+1th iteration position obtained using the second search strategy; UB and LB represent the upper and lower bounds of the variable.

[0080] The third search strategy:

[0081] X3(c+1)=(X best (c)-X M (c))-rand×((UB-LB)×rand+LB)#(6)

[0082] Among them, X3(c+1) represents the c+1th iteration position obtained using the third search strategy.

[0083] The fourth search strategy:

[0084] X4(c+1)=X rand (t)-rand×|X rand (t)-2×rand×X(t)|#(7)

[0085] Wherein, X4(c+1) represents the c+1th iteration position obtained using the fourth search strategy; X rand (t) represents a random selection of a solution from the elite pool.

[0086] Specific Example: The cantilever beam design problem is a structural engineering design problem related to the weight optimization of a cantilever beam with a square cross section. The cantilever beam is rigidly supported at one end and a vertical force acts on the free node of the cantilever. The cantilever beam consists of 5 hollow square blocks with constant thickness, whose height (or width) is the decision variable and the thickness is fixed. The goal is to optimize the parameters of the hollow square section to minimize the weight of the cantilever beam. The model is described in Equation (8)-Equation (11).

[0087] Decision variables

[0088] X=[x1,x2,x3,x4,x5]#(8)

[0089] Objective Function

[0090] 0.0624×(x1+x2+x3+x4+x5)#(9)

[0091] Constraints

[0092]

[0093] Boundary Constraints

[0094] 0.01≤x1,x2,x3,x4,x5≤100#(11)

[0095] The REAOA algorithm is used to solve the cantilever beam design problem and compared with five existing methods. The comparison results are shown in Table 12. Analysis of the experimental results shows that REAOA performs best, with a result of 1.33728120, which is better than the other five comparison algorithms. The comparison results are shown in the following table:

[0096] Table 2 Comparison of results for cantilever beam design problem

[0097]

[0098] From the comparison results in the above table, it can be seen that the REAOA algorithm proposed in the present invention is partially or completely superior to some existing methods in five indicators, and the overall result, that is, the optimal value of the algorithm is better than some existing methods.

[0099] The above embodiments are merely illustrative of the principles and effects of the present invention, and are not intended to limit the present invention. Anyone familiar with the technology may modify or change the above embodiments without violating the spirit and scope of the present invention. Therefore, all equivalent modifications or changes made by those with ordinary knowledge in the technical field without departing from the spirit and technical ideas disclosed by the present invention should still be covered by the claims of the present invention.

Claims

1. A cantilever beam optimization design method, characterized in that: including First, establish a mathematical model for the problem of cantilever beam design; Secondly, introduce it into the Arithmetic Optimization Algorithm (RAOA) of Q-learning; Then, introduce the random elite pool strategy to further optimize the algorithm; Finally, perform iterative solution through the improved arithmetic optimization algorithm; The Arithmetic Optimization Algorithm of Q-learning includes calculating the Q-value of each state-action pair through the Q-learning function given in formula (1), and then saving the Q-values of all state-action pairs in the Q-table: The Arithmetic Optimization Algorithm with Random Elite Pool Strategy includes exploring the solution space through a variety of different search strategies and integrating search strategies with different capabilities.

2. The cantilever beam optimization design method according to claim 1, characterized in that: Among them, the Q-learning function also includes the design of the greedy search parameter ε and the design of the reward function; Design of the greedy search parameter ε: ε linearly increases from 0 to 0.9 in the first c (c < T) iterations, and is then fixed at 0.

9. Set a random number β4 between 0 and 1. When the random number β4 ≥ ε, perform random search. When the random number β4 < ε, perform greedy search; The expression of the reward function is as shown in formula (2): Among them, ω is the adjustment factor, t represents the current number of iterations, and Best t Represents the optimal fitness value of the tth iteration.

3. The optimization design method of a cantilever beam according to claim 2, characterized in that: The steps of the above RAOA algorithm include: Step1: Initialize the state space S, action space A, and Q-table; Step2: Randomly select an action in the initial state; Step3: If β4 > ε, randomly select an action, otherwise select the action with the maximum reward from the Q-table; Step4: Execute the selected action in the AOA algorithm and calculate the reward according to the reward function (2); Step5: Update the Q-table using formula (1); Step6: If the termination condition is met, end the loop, otherwise repeat steps Step3 - Step5.

4. The cantilever beam optimization design method according to claim 1, characterized in that: Among them, the variety of different search strategies in the random elite pool include: The first search strategy: Where rand is a random number between 0 and 1, X1(c+1) represents the c+1th iteration position obtained using the first search strategy, and X best (c) represents the optimal position of the current iteration, X M (c) represents the average of all solutions in the current population.

5. The cantilever beam optimization design method according to claim 4, characterized in that: Among them, the variety of different search strategies in the random elite pool include: The second search strategy: X2(c+1)=(X best (c)-X M (c))×0.1- rand+((UB - LB)×rand+LB)×0.1#(5) Where, X2(c + 1) represents the (c + 1)-th iteration position obtained using the second search strategy, and UB and LB represent the variable upper and lower bounds.

6. The cantilever beam optimization design method according to claim 5, characterized in that: Among them, the variety of different search strategies in the random elite pool include: The third search strategy: X3(c+1)=(X best (c)-X M (c))- rand×((UB - LB)×rand+LB)#(6) Where, X3(c + 1) represents the (c + 1)-th iteration position obtained using the third search strategy.

7. The cantilever beam optimization design method according to claim 6, characterized in that: Among them, the variety of different search strategies in the random elite pool include: The fourth search strategy: X4(c+1)=X rand (t)- rand×|X rand (t)-2×rand×X(t)|#(7) Among them, X4(c+1) represents the c+1th iteration position obtained using the fourth search strategy, and X rand (t) represents a random selection of a solution from the elite pool.

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