Novel meta-heuristic algorithm based on beaver algorithm

Abstract

The invention discloses a novel meta-heuristic algorithm based on a beaver algorithm, and the method comprises the steps: firstly giving an initial density value of each unit, then setting BA operation parameters, and classifying beavers into three types: quality inspection beavers, development beavers and felling beavers; the optimization problem is solved by simulating the group cooperation in the lumbering process so as to solve the optimization model of the topological optimization problem, and then the density is optimized by using BA and iteration is carried out continuously until the optimal optimization result is obtained. The method is used for solving the topology optimization problem, can overcome the limitation of a traditional numerical method in processing a complex optimization problem, improves the optimization efficiency, in addition, in the topology optimization facing challenging engineering problems, BA solution is rapid, the optimization result is stable, no checkerboard phenomenon exists, and compared with other algorithms, the BA has better performance.

Description

Technical Field

[0001] The present invention relates to a topology optimization method, and in particular to a novel meta-heuristic algorithm based on a beaver algorithm (BA). Background Art

[0002] With the continuous advancement of science and technology, the complexity and difficulty of solving optimization problems have increased significantly. Currently, topology optimization faces problems such as unstable solutions, slow solution speeds, and the "checkerboard" phenomenon. Consequently, the Beaver Algorithm, a meta-heuristic algorithm, was proposed to address these complex real-world challenges. During the problem-solving process, the Beaver Algorithm eliminates the need to calculate the gradient of the problem, instead simply calculating the value of the problem under a set of feasible solutions, significantly reducing computational complexity. When solving different problems, only the function to be solved needs to be modified, eliminating the need for extensive modifications to the core algorithm. This simplicity and flexibility have led to the Beaver Algorithm's widespread application in many fields.

[0003] Although new algorithms are constantly being developed, the no free lunch theory states that no specific algorithm can perfectly solve all optimization problems. Therefore, it is necessary to develop different meta-heuristic algorithms to fill the gaps in solving specific problems.

[0004] The present invention proposes a new meta-heuristic algorithm based on population, called Beaver Algorithm (BA). Summary of the Invention

[0005] Purpose of the invention: In view of the shortcomings of the existing technology, the present invention provides a new meta-heuristic algorithm based on the Beaver Algorithm (BA), which further improves the probability of finding the optimal solution of topology to obtain stable and clear topology optimization results.

[0006] Technical solution: A new meta-heuristic algorithm based on the Beaver Algorithm (BA), including the following steps:

[0007] (1) Divide the finite element mesh and establish the optimization model of the topology optimization problem;

[0008] (2) Set the operating parameters of the beaver algorithm and classify beavers into three types: quality inspection beavers, development beavers, and logging beavers;

[0009] (3) Simulate their group cooperation during the logging process to solve the optimization problem, solve the optimization model of the topology optimization problem, and draw the structural topology optimization results.

[0010] In solving multidimensional optimization problems using BA iteration, the parameters n, m, c, maxWalks, stepsize, and maxRaidSteps are first initialized, and the initial population is established. The mathematical formula is shown in Equation (2). Next, the beavers are sorted in ascending order according to their fitness values. The first n beavers in the fitness value ranking are quality inspection beavers, the m beavers with fitness values ​​after the quality inspection beavers are development beavers, and the remaining c beavers are harvesting beavers. The development beavers then walk randomly. The mathematical formulas are shown in Equations (3)-(5). During the walk, if the development beaver's fitness value is better than the quality inspection beaver, it is updated to the quality inspection beaver. If not, the development beaver walks until the maximum number of walks, maxWalks, is reached. Then, the harvesting beaver rushes towards the quality inspection beaver. The mathematical formulas are shown in Equations (6)-(8). During the rush, if the harvesting beaver's fitness value is better than the quality inspection beaver it is rushing, it is updated to the quality inspection beaver. If not, the harvesting beaver rushes until the maximum number of raids, maxRaidSteps, is reached. After reaching the maximum number of raids, the harvester will conduct a local search in that location. The quality inspector will conduct a local search in its location, as shown in equations (9) and (10). During the local search, if the fitness value of the search location is better than the original location, the quality inspector's location is updated to the search location. The optimal topology value is recorded. If the termination condition is met, the optimal solution and optimal value are output. Otherwise, the beavers are rearranged according to their fitness values.

[0011] Initialize the parameters n, m, c, maxWalks, stepsize, and maxRaidSteps, and establish the initial population according to formula (2). Then, sort the beavers in ascending order according to their fitness values. The first n beavers are designated as quality inspection beavers, the m beavers after the quality inspection beavers are designated as development beavers, and the remaining c beavers are designated as harvesting beavers.

[0012] Furthermore, an optimization model for the topology optimization problem is established. In the finite element analysis software, the corresponding unit type is selected to discretize the initial design area into a finite element grid and establish a finite element model. Based on the above finite element model, in the multidimensional optimization problem, each beaver is a column matrix; each column matrix represents a solution to a problem; each variable of the beaver row vector is the density value of a unit. The structural density represented by each row vector is subjected to finite element analysis, and then its density value is updated. In a set of variables (xi, 1, xi, 2, ..., xi, dim), each variable must be between the upper and lower boundaries.

[0013] The initialization of BA is shown in the following formula:

[0014]

[0015] Among them, x is the initial population position, N is the population size, dim is the population dimension, x i，j is the position of individual i in dimension j, xi，j The value of is obtained by the following formula:

[0016]

[0017] in, represents the lower bound of the j-th dimension, represents the upper bound of the j-th dimension.

[0018] Furthermore, the finite element mesh is divided and the optimization model of the topology optimization problem is established as follows: for the optimized structure, the mesh units are divided and the stiffness matrix is ​​assembled to perform finite element analysis. Based on the finite element model, the mathematical model of the topology optimization problem of minimizing the structural flexibility under the volume constraint condition is established as follows:

[0019]

[0020] Where C is the flexibility value, F is the load vector, U is the displacement vector, and V * is the required area, f V is the volume fraction, V i is the area of ​​the i-th unit.

[0021] Furthermore, the development raccoon is specifically: the number of development raccoons is m=2N / 3, the development raccoon randomly walks in the entire timber field, the maximum number of walks is maxWalks, and the mathematical formula of the walk process is as follows:

[0022]

[0023] Among them, stepsize is the walking step length, is a random number in the interval [0, 1].

[0024] Furthermore, during the wandering process, if the amount of wood at the location of the development raccoon is greater than the amount of wood at the original location, the location of the development raccoon is updated to the new location; if the amount of wood at the location of the development raccoon is greater than the amount of wood at the location of the quality inspection raccoon, it is updated to the quality inspection raccoon; the development raccoon repeats the above wandering behavior until the maximum number of wandering times maxWalks is reached.

[0025] Furthermore, the number of harvesting raccoons is c=N / 6. Each harvesting raccoon will select a quality inspection raccoon to raid based on the tournament. The maximum number of raids of the harvesting raccoon is maxRaidSteps. The mathematical formula of the raid process is as follows:

[0026]

[0027] Among them, fit i is the fitness value of the current position of the logging raccoon, fit uThe fitness value of the raiding quality inspection raccoon is selected for the logging raccoon. During the raid, if the amount of wood in the place where the logging raccoon is located is greater than the amount of wood in the place where the quality inspection raccoon is located, the location of the logging raccoon is updated to the location of the quality inspection raccoon. The logging raccoon repeats the above raiding behavior until the maximum number of raids, maxRaidSteps, is reached. After that, the logging raccoon will conduct a local search in the area, and its local search method is the same as that of the quality inspection raccoon.

[0028] Furthermore, the quality inspector beavers compete in each tournament by two quality inspector beavers; in each cycle, quality inspector beavers are randomly selected and their fitness is compared to find the quality inspector beaver with the best fitness (winner); if the fitness of the current tournament winner is higher than that of other quality inspector beavers or the tournament size is equal to the total number of quality inspector beavers, then the index of the winner is returned.

[0029] Furthermore, the number of quality inspection beavers is n=N / 6. The location of the quality inspection beavers is where the largest amount of wood is. They conduct local search in their location. The mathematical formula of the search process is as follows:

[0030]

[0031] Where r1 and r2 are random numbers in the interval [0, 1]; is the velocity vector of particle i in the kth iteration with the dth dimension; is the position vector of particle i in the kth iteration with the dth dimension; is the historical optimal position of particle i in the dth dimension in the kth iteration; is the historical optimal position of the group in the dth dimension in the kth iteration; ω is the inertia factor; c1 is the individual learning factor; and c2 is the group learning factor.

[0032] Furthermore, during the local search process, if the amount of wood at the search location is greater than the amount of wood at the original location of the quality inspection raccoon, the location of the quality inspection raccoon is updated to the search location.

[0033] Furthermore, each beaver in the algorithm represents a topology optimization structure. For each topology optimization structure, the Young's modulus of each element in the stiffness matrix must first be updated according to (12). Then, the stiffness matrix must be reassembled and the displacement value must be solved. After obtaining the displacement value, the fitness fit is obtained according to the calculation formula of (C), thus completing the entire optimization process of the algorithm.

[0034] E=E min +(E max -E min )ρ p (12)

[0035] Among them, E max is the Young's modulus when the material is full, Emin is the minimum Young's modulus, which is calculated to prevent the occurrence of a singular stiffness matrix, ρ is the density value, and p is the penalty coefficient.

[0036] Furthermore, the BA-based topology optimization method according to claim 1 is characterized in that, in the algorithm, the calculation of fitness includes two parts: flexibility value and volume constraint, as described in formula (13). Wherein U0 is the displacement value obtained by finite element analysis when the initial density is given, and this term serves as a regularization term.

[0037]

[0038] Where F is the load vector, U is the displacement vector, and f V is the volume fraction, V i is the area of ​​the i-th unit, ρ i is the density of the i-th unit.

[0039] Compared with the prior art, the present invention has the following beneficial effects:

[0040] 1. The novel metaheuristic algorithm of this invention first establishes an optimization model for the topology optimization problem, then sets the operating parameters of a multi-population genetic algorithm. Finally, it solves the optimization model for the topology optimization problem and plots the structural topology optimization results. By observing beavers' logging behavior, beavers carry out different activities according to their respective divisions of labor and find the optimal solution by sharing and exchanging group information. Furthermore, when setting the operating parameters of the BA, beavers are classified into three types: quality inspection beavers, development beavers, and logging beavers, making the entire algorithm more vivid and concrete.

[0041] 2. In the iterative solution process, the group cooperation of beavers in the logging process is simulated to solve the optimization problem, so as to solve the optimization model of the topology optimization problem and draw the structural topology optimization results. Compared with the existing technology, the method of the present invention can overcome the limitations of traditional numerical methods in dealing with complex optimization problems and improve the efficiency of finding the optimal solution for topology. When using this algorithm to solve the topology optimization problem, finite element analysis is first performed to obtain a preliminary density value, and then the operating parameters of the beaver algorithm are set, and beavers are classified into three types: quality inspection beavers, development beavers, and logging beavers; their group cooperation in the logging process is simulated to solve the optimization problem, so as to solve the optimization model of the topology optimization problem, and then BA is used to optimize the density and continuously iterate until the best optimization result is obtained. In addition, when facing the topology optimization of challenging engineering problems, BA solves quickly, the optimization results are stable, there is no "checkerboard" phenomenon, and it performs better than other algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 This is a schematic diagram of the situation;

[0043] Figure 2 This is a schematic diagram of the final result obtained according to BA in case 1;

[0044] Figure 3 This is a schematic diagram of the final result obtained by the SIMP method according to the comparison algorithm in case 1;

[0045] Figure 4 This is a schematic diagram of situation 2;

[0046] Figure 5 The final result diagram is obtained according to BA for case 2;

[0047] Figure 6 For case 2, the final result diagram is obtained by the SIMP method according to the comparison algorithm;

[0048] Figure 7 This is the flowchart of the BA algorithm. DETAILED DESCRIPTION

[0049] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be further described below.

[0050] Step 1: Divide the finite element mesh and establish an optimization model for the topology optimization problem. In the finite element analysis software, select the corresponding element type to discretize the initial design area into a finite element mesh and establish a finite element model. Based on the above finite element model, in the multidimensional optimization problem, each beaver is a column matrix; each column matrix represents a solution to a problem; in a set of variables (xi, 1, xi, 2, ..., xi, dim), each variable must be between the upper and lower boundaries. The initialization of BA is shown in the following formula:

[0051]

[0052] Where x is the initial population position, N is the population size, dim is the population dimension, and x i，j is the position of individual i in dimension j, x i，j The value of is obtained by the following formula:

[0053]

[0054] in represents the lower bound of the j-th dimension, represents the upper bound of the j-th dimension;

[0055] The method of dividing the finite element mesh and establishing the optimization model for the topology optimization problem is as follows: in the finite element analysis software, the initial design area is discretized into a finite element mesh by selecting a corresponding unit type, and a finite element model is established. Based on the finite element model, a mathematical model for the topology optimization problem of minimizing the structural flexibility under volume constraints is established as follows:

[0056]

[0057] Step 2: Determine the number of various beavers and move them around

[0058] Initialize the parameters n, m, c, maxWalks, stepsize, and maxRaidSteps, and establish the initial population according to formula (2). Then, sort the beavers in ascending order according to their fitness values. The first n beavers are designated as quality inspection beavers, the m beavers after the quality inspection beavers are designated as development beavers, and the remaining c beavers are designated as harvesting beavers.

[0059] (1) Development of raccoon

[0060] Specifically, the number of development raccoons is m = 2N / 3. The development raccoons randomly walk in the entire timber field, and the maximum number of walks is maxWalks. The mathematical formula of the walk process is as follows:

[0061]

[0062] Among them, stepsize is the walking step length, is a random number in the interval [0, 1].

[0063] During the wandering process of the development raccoon, if the amount of wood at the development raccoon is greater than the amount of wood at the original position, the development raccoon's position is updated to the new position; if the amount of wood at the development raccoon's position is greater than the amount of wood at the quality inspection raccoon's position, it is updated to the quality inspection raccoon; the development raccoon repeats the above wandering behavior until the maximum number of wandering times maxWalks is reached.

[0064] (2) Harvesting raccoons

[0065] Specifically, the number of harvester raccoons is c = N / 6. Each harvester raccoon will select a quality inspection raccoon to raid based on the tournament. The maximum number of raids a harvester raccoon can make is maxRaidSteps. The mathematical formula for the raid process is as follows:

[0066]

[0067]

[0068] where fit i is the fitness value of the current position of the logging raccoon, fit uThe fitness value of the raiding quality inspection raccoon is selected for the logging raccoon. During the raid, if the amount of wood in the place where the logging raccoon is located is greater than the amount of wood in the place where the quality inspection raccoon is located, the location of the logging raccoon is updated to the location of the quality inspection raccoon. The logging raccoon repeats the above raiding behavior until the maximum number of raids, maxRaidSteps, is reached. After that, the logging raccoon will conduct a local search in the area, and its local search method is the same as that of the quality inspection raccoon.

[0069] (3) Quality Inspection

[0070] In each tournament, two beavers compete. In each cycle, beavers are randomly selected and their fitness is compared to find the beaver with the best fitness (winner). If the fitness of the current tournament winner is higher than that of other beavers or the tournament size is equal to the total number of beavers, then the index of the winner is returned.

[0071] Specifically, the number of the quality inspection raccoons is n=N / 6. The location of the quality inspection raccoons is where the largest amount of wood is. They conduct local search in the location. The mathematical formula of the search process is as follows:

[0072]

[0073] Where r1 and r2 are random numbers in the interval [0, 1]; is the velocity vector of particle i in the kth iteration with the dth dimension; is the position vector of particle i in the kth iteration with the dth dimension; is the historical optimal position of particle i in the dth dimension in the kth iteration; is the historical optimal position of the group in the dth dimension in the kth iteration; ω is the inertia factor; c1 is the individual learning factor; and c2 is the group learning factor.

[0074] During the local search process, if the amount of wood at the search location is greater than the amount of wood at the original location of the quality inspection raccoon, the location of the quality inspection raccoon is updated to the search location.

[0075] Step 3:

[0076] Record the optimal topology value. If the termination condition is met, output the optimal solution and optimal value. If not, return to step 1.

[0077] To facilitate those skilled in the art to understand the present invention, the present invention is further described below through two specific structural topology optimization examples. For the same example, in addition to the method of the present invention, the SIMP method is also used to solve the problem to demonstrate the advantages of the method of the present invention.

[0078] Example: Optimizing a Cantilever Beam Structure

[0079] The optimization goal of this engineering problem is to find a cantilever beam with the lightest mass and meeting the strength requirements under certain constraints. In this example, E max is 10, E min is 1e-6, and p is 3.

[0080] Case 1:

[0081] The entire structure is divided into 80*40=3200 units, and the variables include the density of each unit. The constraints include the maximum length (a) = 2m, the maximum width (b) = 1m, the Young's modulus E = 10, the volume fraction = 0.3, the external force application point, and the external force (F). The schematic diagram is shown in Figure (1).

[0082] The mathematical model of the problem is as follows:

[0083] The initialization of BA is shown in the following formula:

[0084]

[0085] x i，j =0+rand(0,1)(1-0) (2)

[0086] Initialize the parameters n=12, m=26, c=12, maxWalks=2, stepsize=1.6, maxRaidSteps=2, and make the various beavers start walking.

[0087] The final result obtained by BA is shown in Figure (2), and the comparison algorithm is the SIMP method result as shown in Figure

[0088] (3) The experimental results are shown in the table.

[0089] Case 2:

[0090] The entire structure is divided into 80*40=3200 units, and the variables include the density of each unit. The constraints include the maximum length of the entire structure (a) = 2m, the maximum width (b) = 1m, the Young's modulus E is 10, the volume fraction is 0.3, the external force application point and the external force (F). The schematic diagram is shown in Figure (4):

[0091] The mathematical model of the problem is as follows:

[0092] The initialization of BA is shown in the following formula:

[0093]

[0094] x i，j =0+rand(0,1)(1-0) (2)

[0095] Initialize the parameters n=12, m=26, c=12, maxWalks=2, stepsize=1.6, maxRaidSteps=2, and make the various beavers start walking.

[0096] The final result obtained by BA is shown in Figure (5), and the result of the comparison algorithm is the SIMP method shown in Figure (6). The experimental results are shown in the table.

[0097]

[0098] The results show that BA achieves the best results, which is significantly better than the comparison algorithm, which proves that BA has a good effect in solving the optimization problem of cantilever beam structure.

[0099] The above description is merely a preferred embodiment of the present invention and does not limit the present invention in any way. Any person skilled in the art who, without departing from the scope of the present invention, makes any equivalent substitution, modification, or other changes to the technical solution and technical content disclosed in the present invention shall be deemed to be within the scope of the present invention and still fall within the scope of protection of the present invention.

Claims

1. A new meta-heuristic algorithm called Beaver Algorithm, characterized by: (1) Divide the finite element mesh, give the initial density value of each unit, and establish the optimization model of the topology optimization problem; (2) Set the operating parameters of BA and classify beavers into three types: quality inspection beavers, development beavers, and logging beavers; (3) The group cooperation of beavers in the logging process is simulated to solve the optimization problem in order to obtain the optimization model of the topology optimization problem. Then, BA is used to optimize the density and iterate continuously until the best optimization result is obtained.

2. The novel meta-heuristic algorithm named Beaver Algorithm according to claim 1 is characterized in that: Establish an optimization model for the topology optimization problem. In the finite element analysis software, select the corresponding unit type to discretize the initial design area into a finite element grid and establish a finite element model. Based on the above finite element model, in the multidimensional optimization problem, each beaver is a column matrix; each column matrix represents a solution to a problem; each variable of the beaver row vector is the density value of a unit. For the structural density represented by each row vector, perform finite element analysis and then update its density value. In a set of variables (xi, 1, xi, 2, ..., xi, dim), each variable must be between the upper and lower boundaries. The initialization is shown in the following formula: Among them, x is the initial population position, N is the population size, dim is the population dimension, x i，j is the position of individual i in dimension j, x i，j The value of is obtained by the following formula: in, represents the lower bound of the j-th dimension, represents the upper bound of the j-th dimension.

3. The BA-based topology optimization method according to claim 2, characterized in that: The method of dividing the finite element mesh and establishing the optimization model of the topology optimization problem is as follows: for the optimized structure, the mesh units are divided and the stiffness matrix is ​​assembled to perform finite element analysis. Based on the finite element model, the mathematical model of the topology optimization problem of minimizing the structural flexibility under the volume constraint condition is established as follows: Where C is the flexibility value, F is the load vector, U is the displacement vector, and V * is the required area, f V is the volume fraction, V i is the area of ​​the i-th unit.

4. The novel meta-heuristic algorithm named Beaver Algorithm according to claim 3 is characterized in that: The development raccoons in step (2): the number of development raccoons is m=2N / 3, and the development raccoons randomly wander in the entire timber field, with a maximum number of wandering times of maxWalks. The mathematical formula of the wandering process is as follows: Among them, stepsize is the walking step length, is a random number in the interval [0, 1]; The harvesting raccoons: The number of harvesting raccoons is c = N / 6. Each harvesting raccoon will select a quality inspection raccoon to raid according to the tournament. The maximum number of raids of the harvesting raccoons is maxRaidSteps. The mathematical formula of the raid process is as follows: Among them, fit i is the fitness value of the current position of the logging raccoon, fit u The fitness value of the raiding quality inspector is selected for the harvester. During the raid, if the amount of wood in the place where the harvester is raiding is greater than the amount of wood in the place where the quality inspector is raiding, the harvester's position is updated to the location of the quality inspector. The harvester repeats the above raiding behavior until the maximum number of raids, maxRaidSteps, is reached. After that, the harvester will conduct a local search in the same way as the quality inspector. The quality inspection beavers: The number of quality inspection beavers is n=N / 6. The location of the quality inspection beavers is where the amount of wood is the largest. They conduct local search in their location. The mathematical formula of the search process is as follows: Where r1 and r2 are random numbers in the interval [0, 1]; is the velocity vector of particle i in the kth iteration with the dth dimension; is the position vector of particle i in the kth iteration with the dth dimension; is the historical optimal position of particle i in the dth dimension in the kth iteration; is the historical optimal position of the group in the dth dimension in the kth iteration; ω is the inertia factor; c1 is the individual learning factor; and c2 is the group learning factor.

5. The BA-based topology optimization method according to claim 4, characterized in that: During the wandering process of the development raccoon, if the amount of wood at the development raccoon is greater than the amount of wood at the original position, the development raccoon's position is updated to the new position; if the amount of wood at the development raccoon's position is greater than the amount of wood at the quality inspection raccoon's position, it is updated to the quality inspection raccoon; the development raccoon repeats the above wandering behavior until the maximum number of wandering times maxWalks is reached.

6. The BA-based topology optimization method according to claim 4, characterized in that: The quality checkers compete in each tournament between two quality checkers. In each round, a quality checker is randomly selected and their fitness is compared to find the quality checker with the best fitness (winner). If the fitness of the current tournament winner is higher than that of other quality checkers or the tournament size is equal to the total number of quality checkers, then the index of the winner is returned.

7. The BA-based topology optimization method according to claim 4, characterized in that: During the local search process, if the amount of wood at the search location is greater than the amount of wood at the original location of the quality inspection raccoon, the location of the quality inspection raccoon is updated to the search location.

8. The BA-based topology optimization method according to claim 7, characterized in that: Each beaver in the algorithm represents a topology optimization structure. For each topology optimization structure, it is first necessary to update the Young's modulus of each unit in the stiffness matrix according to (E's formula), then reassemble the stiffness matrix, solve the displacement value, and after obtaining the displacement value, obtain the fitness fit according to (C's calculation formula), thus completing the entire optimization process of the algorithm. E=E min +(And max -AND min )ρ p (12) Among them, F max is the Young's modulus when the material is full, E min is the minimum Young's modulus, which is calculated to prevent the occurrence of a singular stiffness matrix, ρ is the density value, and ρ is the penalty coefficient.

9. The BA-based topology optimization method according to claim 8, characterized in that: In the algorithm, the calculation of fitness includes two parts: flexibility value and volume constraint, as described in the following formula, where U0 is the displacement value obtained by finite element analysis when the initial density is given, and this term serves as a regularization term. Where F is the load vector, U is the displacement vector, and f V is the volume fraction, V i is the area of ​​the i-th unit, ρ i is the density of the i-th unit.

Images