Transonic flow resistance optimization method for RAE 2822 airfoil using an improved sparrow search method based on penalty function constraint optimization

Abstract

The present invention relates to an improved sparrow search method based on penalty function constraint optimization. This method is an improved sparrow search method based on penalty function constraint optimization that integrates the sine-cosine algorithm and the light-sensing strategy of fireflies. By generating the initial population in combination with Cubic chaotic mapping in the initial population initialization stage, the accuracy and distribution of the initial population are improved; in the iterative process of updating the position of the followers, the sine-cosine algorithm is added to improve the position update strategy of the followers, making the search mechanism of the followers more flexible, expanding the global exploration range, increasing the population diversity, increasing the probability of finding the optimal solution, and accelerating the convergence speed; finally, the light-sensing idea of the firefly algorithm is used to refine and optimize the method, helping the method to jump out of local extrema, providing a better parental population for the next iteration, providing an optimization direction for the method, avoiding blind search, accelerating convergence, and thus improving the optimization quality of the method.

Description

Technical Field

[0001] The present invention belongs to swarm intelligence optimization algorithms, and relates to a transonic flow resistance optimization method for the RAE 2822 airfoil using an improved sparrow search method based on penalty function constraint optimization. Background Art

[0002] The optimization problem (Optimization Problem) is to solve the maximum or minimum value of the objective function under certain constraint conditions. The constrained optimization problem is an important branch in this field. The constraint conditions include but are not limited to: upper and lower bound constraints of decision variables, equality and inequality constraints. The definition formula of the constrained problem is as follows:

[0003]

[0004] where g l is the constraint imposed on f, and the n variables take values between z i and o i . The solution obtained is represented as a vector x, and this vector needs to satisfy the constraints therein while making f obtain the optimal solution. This requires that when the algorithm solves, multiple sets of vector solutions x k ,…,x k+1 need to be solved, where k = 1,…, n - 1. For x k and x k+1 , if the number of elements in x i that satisfy formula (1) is more than the number of elements in x k+1 that satisfy formula (1), then the former is better than the latter; if the number of elements in the two vectors that satisfy formula (1) is the same, then compare the fitness functions of x k and x k+1 , and take the better one as the better solution. In summary, the number of x k that satisfy the inequality g l (x 1,..., x i ) ≧ y l is the first evaluation criterion, and the second evaluation criterion is the fitness value of x k .

[0005] There are challenges in solving constrained problems such as discrete constraint regions, equality constraints, and non-linear constraints, which will all be transformed into unconstrained problems for processing in the actual solution process. The main research goal of constrained optimization evolutionary algorithms is to use constraint handling techniques and evolutionary algorithms to solve constrained optimization problems. Currently, the mainstream constraint handling techniques are divided into six categories: penalty function method, ε-constraint handling method, feasibility rule, stochastic ranking method, and multi-objective optimization. These methods each have their own advantages and disadvantages.

[0006] In the book *Adaption in Natural and Artificial System* published by Professor John Holland of the University of Michigan in 1975, the adaptive change mechanisms in intelligent systems and nature were elaborated, which is regarded as the pioneering work of swarm intelligence algorithms. This type of algorithm is developed by studying the group behaviors of social animals or insects and has good performance in optimization problems that are difficult to solve directly. A heuristic algorithm is an algorithm constructed by applying real-world experience or human intuitive ideas within limited time and space costs, and it can obtain as many feasible solutions as possible through algorithmic deduction for the target problem. As a type of heuristic algorithm, the swarm intelligence algorithm maps natural behaviors such as predation and evolution of group activities in nature to the retrieval and optimization of the algorithm in the search area of the problem, maps the points retrieved within the given time and space range to the position information of individual group animals during activities, and maps the solution of the function of the target problem to be solved to the adaptability of the individual in the environment.

[0007] In the past decade or so, a variety of bionic swarm intelligence algorithms have been proposed in the academic community. They all have strong randomness and excellent retrieval capabilities, but each algorithm has certain advantages and disadvantages. For example, the Particle Swarm Optimization (PSO) algorithm has a low time complexity and fast retrieval, but its search ability decreases in the later stage of iteration and it is prone to premature convergence; the Artificial Bee Colony Algorithm (ABC) has good global search performance, but the population diversity is insufficient in the later stage of iteration; the Ant Colony Optimization (ACO) algorithm strengthens the global search by adding a positive feedback mechanism, but the disadvantage is that the convergence speed is slow and it is also easy to fall into the local optimum; the Firefly Algorithm (FA) has few parameter settings, and the application of the light sense idea makes the individual converge to the local optimum, so it has strong local search ability, but the speed is not fast and the accuracy is also insufficient; the Grey Wolf Algorithm (GWO) also has few parameters and a simple process, and its retrieval ability is also excellent, but the population diversity is not rich enough; the Whale Optimization Algorithm (WOA) has the characteristic of high scalability, but its disadvantages are similar to those of the ant colony optimization algorithm.

[0008] The Sparrow Search Algorithm (SSA) is a swarm intelligence optimization algorithm proposed in 2020, inspired by the foraging behavior of sparrows in groups and their alertness and avoidance behavior towards natural enemies. It has excellent optimization ability, fewer iteration times for convergence, and incorporates a reconnaissance and early warning mechanism. Since this algorithm has fewer control parameters and a relatively simple implementation process, it has certain significance in solving problems such as evacuation route planning, traveling salesman problem, and image segmentation. At the same time, in practical applications, some scholars have proposed methods for planning the flight trajectory of unmanned aerial vehicles, detecting network intrusion behavior, detecting faults in fiber optic gyroscopes, and optimizing lighting control based on the Sparrow Search Algorithm.

[0009] In terms of its overall structure, the Sparrow Search Algorithm is similar to the Artificial Bee Colony Algorithm. It has a certain dependence on the initial solution, and the iterative process has an adverse effect on the diversity of the population. The most significant problem is that it is prone to falling into local optima and difficult to jump out, with relatively weak global search ability. To improve its performance, research on this algorithm has been carried out both at home and abroad. The main directions include: optimizing the population initialization method, optimizing the iterative process of updating individual positions, and superimposing mutation operators. For example, in an improved Sparrow Search Algorithm that combines multiple strategies such as K-Mean clustering, the initial population is processed by K-Mean clustering, and the idea of the sine-cosine algorithm and the adaptive local search strategy are combined to improve the convergence accuracy and optimization ability; another proposed method is to fuse multiple strategies into the Sparrow Search Algorithm. A chaotic reverse learning strategy is added to improve the ability to jump out of local optima, and a certain balance is maintained between local and global searches using the Chicken Swarm Optimization (CSO). Finally, the Cauchy-Gaussian mutation strategy is adopted to maintain diversity and improve the anti-stagnation ability. In their respective literatures, both improved algorithms have experimentally proven that they have better performance compared to the original Sparrow Search Algorithm. Summary of the Invention

[0010] The purpose of the present invention is to provide a method for optimizing the transonic flow resistance of the RAE 2822 airfoil using an improved Sparrow Search method based on penalty function constraint optimization, aiming at the shortcomings of the Sparrow Search Algorithm.

[0011] To achieve the above purpose, the technical solution of the present invention is: A method for optimizing the transonic flow resistance of the RAE 2822 airfoil using an improved Sparrow Search method based on penalty function constraint optimization, comprising the following steps:

[0012] S1. In the initial population stage, optimize the initial sparrow population by combining Cubic chaotic mapping;

[0013] S2. During the iterative process of follower position update, optimize the follower position update strategy by combining the sine-cosine algorithm strategy;

[0014] S3. Combine the firefly algorithm perturbation optimization for refinement, expand the search range, retain better individuals for the next generation, and provide a search direction;

[0015] S4. Use the penalty function method to achieve the transformation of the objective optimization problem from a constrained problem to an unconstrained problem in order to obtain the solution of the constrained problem.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] 1. Optimize the initial sparrow population by combining the Cubic chaotic map to disperse the population and improve the diversity of the population.

[0018] 2. Optimize the strategy for updating the follower position during the iterative process by combining the sine-cosine algorithm strategy, making the follower search mechanism more flexible, better adjusting the balance between the global exploration ability and the local development ability, enhancing the ability to jump out of the local optimal trap, and optimizing the optimization performance.

[0019] 3. Combine the firefly algorithm perturbation optimization for refinement, expand the search range, retain better individuals for the next generation, provide a search direction, strengthen the purposefulness of the population search, and improve the convergence speed and accuracy of the solution set.

[0020] 4. Use the penalty function method to achieve the transformation of the objective optimization problem from a constrained problem to an unconstrained problem in order to obtain the solution of the constrained problem. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is the flow chart of the sparrow search method.

[0022] Figure 2 It is the convergence curve of the average best drag coefficient obtained by ISSMPF, SSA, SCA, and FA. DETAILED DESCRIPTION OF THE INVENTION

[0023] The following will specifically describe the technical solution of the present invention with reference to the drawings.

[0024] A transonic flow drag optimization method for the RAE 2822 airfoil using an improved sparrow search method based on penalty function constraint optimization of the present invention includes the following steps:

[0025] S1. Optimize the initial sparrow population by combining the Cubic chaotic map during the initialization population stage;

[0026] S2. During the iterative process of follower position update, optimize the follower position update strategy by combining the sine-cosine algorithm strategy;

[0027] S3. Refine and optimize by combining the firefly algorithm perturbation search to expand the search range, retain better individuals for the next generation, and provide a search direction;

[0028] S4. Use the penalty function method to realize the transformation of the objective optimization problem from a constrained problem to an unconstrained problem in order to obtain the solution of the constrained problem.

[0029] The following is the specific implementation process of the present invention.

[0030] 1. Sparrow Search Algorithm

[0031] The initial version of the sparrow search algorithm stipulates that the initial population includes three types of "sparrows", that is, individuals within the domain, which are respectively discoverers, followers, and scouts. The position of the sparrow is X i =(x i1 ,x i2 ,...,x iD ), i = 1, 2,..., N, the fitness value is f = (f1, f2,..., f N ), D is the dimension of the space where the sparrow is located, and N is the population individuals. The population of N sparrows is first sorted according to the fitness quality, and the first P individuals with high fitness are selected to form discoverers. The remaining N - P individuals are called followers. In addition, a certain proportion of individuals are determined from the entire population to form scouts aware of the threat of natural enemies. Usually, discoverers can provide a reference basis for the foraging range and orientation for followers due to having better fitness values. Followers will follow the discoverers for better food (that is, a better solution to the problem) to ensure the predation rate. When scouts discover predators, they send out alarm signals, and the sparrow population makes anti-predation behaviors.

[0032] The position update formula for discoverers is as follows:

[0033]

[0034] In formula (2), the iteration number is represented by t, and the maximum iteration number is represented by T; X t ij represents the j-th dimension of the i-th individual in the t-th iteration; ɑ is a random number in the range (0, 1], R2 is the early warning value, set in the range [0, 1], Q is a random number and follows a Gaussian distribution, and ST is the safety value. L is a 1×d matrix with all elements being 1. When R2 < ST, there are no predators in the foraging environment of the sparrows as discoverers, and these individuals can perform a wider search for foraging; when R2 ≥ ST, a certain proportion of scouts in the population have discovered predators, and the discoverers immediately move to a safer area to continue the subsequent foraging.

[0035] The formula based on which the follower position is updated is:

[0036]

[0037] In formula (3), the position with the optimal fitness where the discoverer individual is located is represented by X P represents, and X worst represents the position with the least satisfactory fitness within the foraging area. A is a matrix, which is 1×d, and all elements in it are randomly assigned as 1 or -1, and satisfy A + = A T (AA T ) -1 . In the case of i > n / 2, the followers with lower fitness values will use the exp function to move to the area with higher fitness values. In other cases, the followers will actively move closer to the discoverer's position to forage better.

[0038] The formula for the anti-predation behavior of sparrows when the scout realizes danger is:

[0039]

[0040] In formula (4), the best position within the entire foraging range (i.e., the domain) is represented by the parameter X best represents, β is the step size control parameter that follows the normal distribution of σ 2 = 1, μ = 0, K is a random number on [-1, 1], which is used to represent the direction of the sparrow individual to update its position. In addition, the fitness value of sparrow i is represented by f i represents, the worst fitness value in the entire domain is represented by f w represents (worst), f g is the best, and ε is a small constant used to ensure that the denominator is not zero. f i > f g describes that the position of the sparrow is relatively remote and is at risk of being attacked by natural enemies, that is, there is a risk of generating an unavailable solution; f i＝ f g describes that the sparrows in the relatively dense population distribution area realize the dangerous situation and choose to group with other sparrow individuals in the population to avoid risks, so they move closer to other individuals.

[0041] In summary, the general steps of the sparrow search algorithm are as follows: After initializing the population, divide the individuals into two types, discoverers and followers, and only these two types of identities. Then, sequentially update the positions of the two types of identity individuals. At the same time, some individuals realize the danger and become scouts, and their positions will be updated again. After the loop iteration process is determined to end, the final result is obtained. The process of the sparrow search algorithm is as Figure 1 shown.

[0042] 2. Improved Sparrow Search Method Based on Penalty Function Constraint Optimization

[0043] 2.1. Introduction of Cubic chaotic mapping

[0044] To a certain extent, the quality and diversity of the initial population in swarm intelligence optimization algorithms will affect the accuracy of the solution. A high-quality initial population can provide the direction for the algorithm to search for the global optimal solution and improve the convergence speed and accuracy of the algorithm. The initial population generated by the standard sparrow search algorithm using the random initialization method is prone to poor population diversity and uneven distribution. Therefore, the initialization of the sparrow population has a great impact on the search accuracy of the sparrow search algorithm. In order to obtain a more widely distributed population within the domain of the problem and enhance population diversity, the present invention introduces Cubic chaotic mapping to initialize the population. The basic expression of the Cubic mapping is as follows:

[0045]

[0046] In formula (5), b and c are chaotic influence factors, and the values of these two parameters will directly affect the range of the Cubic mapping. Generally, when c ∈ (2.3, 3), the sequence generated by this mapping is considered to be in a chaotic state. Additionally, when b = 1, x n ∈ (-2, 2); when b = 4, x n ∈ (-1, 1). Scholars have carried out a series of calculations on the maximum Lyapunov exponents of 16 one-dimensional chaotic mappings. Among them, the Cubic mapping expression is obtained as:

[0047]

[0048] In formula (6), x n ∈ (-1, 1), and ρ, as a control parameter, greatly affects the chaotic nature of the sequence of operation results of the Cubic mapping function. The results in relevant literature show that the maximum Lyapunov exponents of the Logistic mapping, Tent mapping, and Cubic mapping are similar, and their performance is better than that of the Sine mapping, Singer mapping, etc. Therefore, the Cubic mapping is selected to improve the strategy of population initialization.

[0049] 2.2. Incorporation of the position update of the sine-cosine algorithm

[0050] The Sine Cosine Algorithm (SCA) was proposed by scholar Seyedali Mirjalili in 2016. It is a heuristic optimization algorithm that uses sine and cosine functions to design mathematical formulas. The algorithm fluctuates outward based on the mathematical model of sine and cosine or in the direction of the optimal solution, and uses multiple random variables and adaptive variables to calculate the position of the current solution, so as to search different regions in the search space, effectively avoid local optima, and converge to the global optimum. In the standard Sparrow Search Algorithm, the update of the follower's position largely depends on the guidance of the discoverer, which limits the search direction and range of the follower. Once the discoverer falls into a local optimum, the follower is also likely to fall into the local optimum and it is difficult to get out, and the performance of the overall algorithm will also decline. In addition, the proportion of discoverers remains constant throughout the search process. When a follower discovers a better solution, it can be promoted to a discoverer, and the worst of the original discoverers will be demoted to a follower. Therefore, it is particularly important to improve the optimization ability of the follower, that is, to improve the optimization ability of the discoverer to a certain extent. After introducing the Sine Cosine Algorithm formula into the follower's updated position iteration, it can not only reduce the dependence of the follower's updated position on the discoverer, but also expand the search range of the follower and improve the diversity of the follower, thus avoiding the low optimization efficiency and premature convergence of the algorithm caused by the population being overly concentrated on local optimal points. The update calculation formula of the Sine Cosine Algorithm is as follows:

[0051]

[0052] In formula (7), the position of individual i at iteration number t is represented by The position of the optimal individual is represented by In the expression, t represents the current update number, i represents the number of individuals, the value range of r2 is [0, 2π], both r3 and r4 are randomly distributed numbers with an average in the range of [0, 1], and the calculation of r1 is based on the following formula:

[0053]

[0054] In formula (8), the value of a in the present invention is 2, t represents the current iteration number, and T represents the maximum iteration number; as the iteration number increases, r1 will also increase. In the stage where r1 > 1, the algorithm has a stronger exploration ability; in the subsequent stage where r1 < 1, the algorithm has a stronger exploitation ability. The parameter r2 affects the step size when the individual brought into the algorithm enters the next iteration, and the parameter r3 can affect the degree of dependence of the algorithm on the optimal solution individual during the iteration process. When r3 < 1, the degree of dependence on the optimal solution individual is increased, and when r3 > 1, the dependence degree is decreased. r4 ensures that the switching between the two formulas of the algorithm is equiprobable.

[0055] 2.3. Refinement Strategy Based on Firefly Algorithm

[0056] The firefly algorithm was proposed by scholars such as Yang X.S. in 2008 and was improved by combining the Levy flight algorithm strategy two years later. It is an algorithm with relatively rich development among the swarm intelligence algorithms proposed in the 2000s. The inspiration for the firefly algorithm comes from the luminous behavior within the firefly population, mimicking the mutual information transmission and degree of attraction among individuals within the population. In the algorithm, it is stipulated that each individual will attract each other, only referring to two parameters: the brightness of the fluorescence and the distance between individuals, and gender is not within the scope of consideration. The brighter firefly attracts the darker firefly. In addition, the light intensity I is inversely proportional to the square of the distance r from the light source and follows formula (9).

[0057]

[0058] This limits the visibility of individuals within a finite radius. In the entire space, the individual with the highest visible fluorescence brightness among all individuals in the population randomly updates its position in the space. The distance is r, and the formula for calculating the brightness of one firefly relative to another is as follows:

[0059]

[0060] In formula (10), I0 represents the brightness of the firefly itself at this position, and γ is an artificially set absorption brightness factor used to simulate the influence of the medium in the air on the light intensity, usually set to 1, and r ij represents the Euclidean distance between the firefly individuals at positions i and j respectively, and this item is calculated according to formula (11).

[0061] r ij =||x i -x j || (11)

[0062] We can assume that the brightness of firefly individual i is greater than the brightness of firefly individual j itself. According to the algorithm rules, firefly i will attract firefly j, and then firefly i will cause firefly j to actively approach it. The formula (12) for calculating the position update information of firefly j is as follows:

[0063] x j (t + 1)=x j (t)+β ij (x i (t)-x j (t))+αε j (12)

[0064] In formula (12), x j (t) is the position of firefly j in the t-th iteration, and x j(t + 1) is the position of firefly j at the (t + 1)-th iteration of approaching firefly i this time. The parameter α represents the step size factor, satisfying α ∈ (0, 1), and the parameter ε j is a random number that follows a normal distribution on [0, 1]. The attraction of individual i to individual j is represented by β ij and its calculation depends on formula (13).

[0065]

[0066] In formula (13), β0 represents a manually specified attraction value of an individual to an individual with a distance of 0 from itself.

[0067] In the firefly algorithm, firefly individuals are randomly distributed. Each individual has its own perception ability and searches for the brightest individual within its own search radius area and moves towards it. Therefore, the randomness idea of each individual's independent search in the firefly algorithm is used to refine the solution of the sparrow search algorithm, retain better offspring for the next iteration, clarify the search direction, reduce blind search, further ensure the diversity of the population, weaken the interference of local extrema, and improve the convergence speed and accuracy.

[0068] 2.4. Constraint Handling Based on Penalty Function

[0069] The present invention uses the penalty function method. The main solution idea of this method is to transform the target optimization problem from a constrained problem to a series of unconstrained problems to obtain the solution of the original linear constrained problem. Because when it is transformed into solving an unconstrained problem, the algorithm will not consider whether the obtained optimal individual meets the constraint conditions during the original operation stage, so a penalty function is added as a penalty term. As the number of population position updates increases, the requirements of the penalty term become more stringent and the penalty amount increases, thus forcing the population to approach the area specified by the constraints to become individuals that meet the conditions, which are the feasible solutions of the original optimization problem. A major feature of the penalty function method is that it takes into account the richness of population individuals, has a relatively wide distribution within the domain, has no restrictions on the distribution state of the initial population, and is restricted during the iteration process of the swarm intelligence algorithm. Therefore, it is more suitable for the improved sparrow search algorithm of the present invention. The quadratic penalty function method in the penalty function method is adopted in the present invention. This method stipulates a quadratic penalty function Q(x, μ), and its mathematical expression is as follows:

[0070]

[0071] In formula (14), a penalty parameter μ not less than 0 is set, and the value of μ refers to the penalty sequence {μ k} columns, when the population iterates, the value of k will continuously increase, the value of μ will continuously decrease, and 1 / (2μ) will increase very rapidly to conform to the theoretical idea of the penalty function method. When k approaches positive infinity and μ approaches 0, the approximate minimum value Q(x k , μ k ) is obtained. Since one of the terms used for punishment in this polynomial ( is a quadratic function, it is differentiable. Thus, at the approximate minimum point x k and x k-1 near it, etc., can be regarded as the initial points with better quality for the (k + 1)-th update.

[0072] 2.5. Algorithm Flow

[0073] The flow of the improved sparrow search method based on penalty function constraint optimization (ISSMPF) is shown in the following steps:

[0074] Step 1: Set basic parameters such as the warning value ST, the proportion of discoverers PD, the proportion of sparrows aware of danger SD, the number of discoverers PDNumber, etc.

[0075] Step 2: Initialize the population using the Cubic mapping according to formula (6), and calculate the initial fitness value of the population according to formula (14), and sort to obtain the optimal fitness value G Best and the global optimal position X Best .

[0076] Step 3: Update the position of the discoverers according to formula (2).

[0077] Step 4: Update the position of the followers according to formula (3), and then perform the sine-cosine operation according to formula (7).

[0078] Step 5: Randomly select scouts and update their positions according to formula (4).

[0079] Step 6: Perform perturbation optimization on the sparrows in the population using the firefly light perception strategy according to formula (12).

[0080] Step 7: Calculate the population fitness value according to formula (14) and update G Best and X Best .

[0081] Step 8: Judge whether the termination condition is satisfied; if yes, end and return the optimal solution; otherwise, return to Step 3 and repeat Steps 3 - 8.

[0082] 2.6. Practical Application

[0083] In this section, the ISSMPF is used to solve an actual expensive constrained optimization problem: minimizing the drag of the transonic flow over the RAE 2822 airfoil. Among them, time-consuming computational fluid dynamics (CFD) simulations are used to evaluate the airfoil performance (i.e., the airfoil obtained after algorithm optimization). The present invention converts the airfoil shape parameters into 18 design variables through the shape classification function conversion method, where the range of each design variable is [0, 1]. Without loss of generality, the mathematical formula for this problem is as follows:

[0084]

[0085] where c d (x), c l (x), c m (x) and Area(x) represent the drag coefficient, lift coefficient, pitching moment, and airfoil area, respectively.

[0086] Since it takes about 18 minutes to simulate a transonic airfoil using PMNS2D (CFD code) in the present invention, in order to save computational costs, the number of simulations is set to 500. Figure 2 The convergence curves of the average best drag coefficients obtained by ISSMPF, SSA, SCA, and FA are shown. From Figure 2 it can be found that the performance of ISSMPF is the best. This shows the effectiveness and superiority of ISSMPF when solving RAE2822.

[0087] The above are the preferred embodiments of the present invention. All changes made according to the technical solution of the present invention, when the functions and effects produced do not exceed the scope of the technical solution of the present invention, shall fall within the protection scope of the present invention.

Claims

1. An optimization method for the transonic flow resistance of the RAE 2822 airfoil using an improved sparrow search method based on penalty function constraint optimization, characterized in that It includes the following steps: S1. Optimize the initial sparrow population by combining Cubic chaotic mapping in the initial population stage; S2. Optimize the follower position update strategy by combining the sine-cosine algorithm strategy during the iterative process of follower position update; S3. Refine and optimize by combining the perturbation optimization of the firefly algorithm to expand the search range, retain better individuals for the next generation, and provide a search direction; S4. Use the penalty function method to realize the transformation of the objective optimization problem from a constrained problem to an unconstrained problem to obtain the solution of the constrained problem; This method is applied to the drag minimization optimization of transonic flow on the RAE 2822 airfoil. Specifically, the airfoil shape parameters are converted into 18 design variables through the shape classification function conversion method. Among them, the range of each design variable is [0,1]. The mathematical formula for the drag minimization of transonic flow on the RAE 2822 airfoil is as follows: Minimize: c d (x) Subject to:c l (x)≥0.824 c m (x) ≥ -0.092 Area(x)≥Area initial Among them, c d (x), c l (x), c m (x) and Area(x) represent the drag coefficient, lift coefficient, pitching moment, and airfoil area respectively; The specific implementation of step S2 is as follows: During the iterative process of follower position update in the sparrow search method, the sine-cosine algorithm is introduced. The update calculation formula of the sine-cosine algorithm is as follows: where the position of individual \(i\) at iteration \(t\) is represented by and the position of the optimal individual is represented by where \(r_2\) ranges from \([0, 2\pi]\), \(r_3\) and \(r_4\) are both uniformly distributed random numbers in the range \([0, 1]\), and the calculation formula for \(r_1\) is as follows: In the formula, a takes the value of 2, t represents the current iteration number, and T represents the maximum iteration number; as the iteration number increases, r1 will also increase; in the stage where r1>1, the sine-cosine algorithm has stronger exploration ability; then in the stage where r1<1, the sine-cosine algorithm has stronger development ability. The parameter r2 affects the step size when the individual brought into the sine-cosine algorithm enters the next iteration. The parameter r3 affects the degree of dependence on the optimal solution individual during the iteration of the sine-cosine algorithm. When r3<1, the degree of dependence on the optimal solution individual is increased, and when r3>1, the dependence degree is decreased. r4 ensures that the switching between the two formulas of the sine-cosine algorithm is equiprobable; The specific implementation method of step S4 is: use the penalty function method to transform the target optimization problem from a constrained problem to a series of unconstrained problems to obtain the solution of the original linear constrained problem. The specific implementation is as follows: Adopt the quadratic penalty function method in the penalty function method, and define a quadratic penalty function Q(x,μ), and its mathematical expression is as follows: In the formula, a penalty parameter μ not less than 0 is set, and the value of μ refers to the penalty sequence {μ k}. When the population iterates, the value of k will continuously increase, and the value of μ will continuously decrease. 1 / (2μ) will increase very rapidly to conform to the theoretical idea of the penalty function method; when k approaches positive infinity and μ approaches 0, the approximate minimum value Q(x k , μ k ) of the penalty function is obtained; since one term used for penalty in the polynomial is a quadratic function, it is differentiable; thus, at the approximate minimum point x k and its nearby x k-1 positions are regarded as the initial points with good quality for the (k + 1)-th update.

2. The transonic flow resistance optimization method of the RAE 2822 airfoil using the improved sparrow search method based on penalty function constraint optimization according to claim 1, characterized in that, The specific implementation of the said step S1 is as follows: In the initial population stage of the sparrow search method, Cubic chaotic mapping is introduced to initialize the population. The expression of the Cubic mapping is as follows: where x n ∈ (-1, 1), and ρ, as a control parameter, greatly affects the chaotic property of the result sequence of the Cubic mapping function operation.

3. The transonic flow resistance optimization method of the RAE2822 airfoil using the improved sparrow search method based on penalty function constraint optimization according to claim 1, characterized in that, The specific implementation method of step S3 is: utilize the randomness idea of independent search of each individual in the firefly algorithm to refine and optimize the solution of the sparrow search algorithm, retain better offspring for the next iteration, clarify the search direction, reduce blind search while ensuring the diversity of the population, weaken the interference of local extrema, and improve the convergence speed and accuracy.

4. The transonic flow resistance optimization method of the RAE2822 airfoil using the improved sparrow search method based on penalty function constraint optimization according to claim 1, characterized in that, The specific implementation steps of this method are as follows: Step 1. Set the basic parameters of the warning value ST, discoverer ratio PD, sparrow ratio SD aware of danger, and discoverer number PDNumber; Step 2. Initialize the population using Cubic mapping according to formula (1), calculate the initial fitness value of the population according to formula (2), and sort to obtain the optimal fitness value GBest and the global optimal position XBest; In formula (1), x n ∈ (-1, 1), and ρ is a control parameter, which greatly affects the chaos of the operation result sequence of the Cubic mapping function; Step 3. Update the discoverer position according to formula (3); In formula (3), the number of iterations is represented by t, and the maximum number of iterations is represented by T; X t ij represents the j-th dimension of the i-th individual in the t-th iteration; ɑ is a random number in the range (0, 1], R2 is the warning value, set in the range [0, 1], Q is a random number and follows a Gaussian distribution, ST is the safety value, and L is a 1×d matrix with all elements equal to 1; when R2 < ST, there are no predators in the foraging environment of the sparrows acting as discoverers, and these individuals perform a wider search during foraging; when R2 ≥ ST, a predetermined proportion of the scouts in the population discover predators, and the discoverers immediately move to a safer area and continue foraging later; Step 4: Update the follower's position according to Equation (4), and then perform sine-cosine operations according to Equation (5); In formula (4), the position with the optimal fitness where the discoverer individual is located is represented by X P represents, and X worst represents the position with the least satisfactory fitness in the foraging area. A is a matrix, which is 1×d, and all elements in it are randomly assigned 1 or -1, and satisfy A + =A T (AA T ) -1 ; in the case of i > n / 2, the followers with low fitness values will use the exp function to move to the area with higher fitness values. In other cases, the followers actively move closer to the discoverer's position to forage better; Step 5: Select the scout according to Equation (6) and update the position; In Equation (6), the full predation range, i.e., the optimal position within the domain, is represented by the parameter X best represents, and β is the step size control parameter that follows a normal distribution with σ 2 = 1 and μ = 0. K is a random number on [-1, 1] used to represent the direction of the sparrow individual to update its position. Additionally, the fitness value of sparrow i is represented by f i represents, the worst fitness value in the entire domain is represented by f w represents (worst), f g is the best, and ε is a small constant used to ensure that the denominator is not zero; f i > f g describes that the position of the sparrow is relatively remote and is prone to the risk of being attacked by natural enemies, that is, there is a risk of generating an unavailable solution; f i = f g describes that the sparrows in the denser part of the population distribution are aware of the dangerous situation and choose to group with other sparrow individuals in the population to avoid risks, so they get closer to other individuals; Step 6: Perform disturbance optimization on the firefly light-sensing strategy for the sparrows in the population according to Equation (7); x j (t + 1) = x j (t) + β ij (x i (t) - x j (t)) + αε j (7) In Equation (8), x j (t) is the position of firefly j in the t-th iteration, and x j (t + 1) is the position of firefly j in the (t + 1)-th iteration when approaching firefly i in this time. The parameter α represents the step size factor, satisfying α ∈ (0, 1), and the parameter ε j is a random number that satisfies the normal distribution on [0, 1]; the attraction of individual i to individual j is represented by β ij , and its calculation depends on Equation (8): In Equation (8), β0 represents a manually specified attraction value of an individual to an individual with a distance of 0 from itself; Step 7: Calculate the population fitness value according to Equation (2) and update GBest and XBest; Step 8: Determine whether the termination condition is satisfied; if so, end and return the optimal solution; otherwise, return to Step 3 and repeat Steps 3-8.

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