An electromagnetic metasurface unit structure optimization method based on constraint space mapping

Abstract

The application discloses a kind of electromagnetic super surface unit structure optimization method based on constraint space mapping, belong to electromagnetic super material design technical field.The application is to solve the problem of low optimization efficiency and difficult to handle constraint condition in the process of super surface unit structure optimization in prior art, mainly using mapping multidimensional structure variable to the convex constraint space formed by inequality constraint condition, and combining multi-objective optimization algorithm for efficient optimization.By using direction vector, scale factor to represent structure variable vector, the initial solution generation process that satisfies complex constraints is simplified, and the global search ability and convergence speed in the constraint space are improved.The method can quickly converge to local optimal solution under complex constraint conditions, and effectively find the Pareto optimal solution, thereby providing an efficient and accurate solution for multi-objective optimization of electromagnetic super surface unit structure.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic metamaterial design technology, specifically a method for optimizing the structure of electromagnetic metasurface units based on constrained space mapping. Background Technology

[0002] Electromagnetic metasurfaces are two-dimensional subwavelength structures with different geometries. By appropriately designing the unit cell structure, the amplitude and phase of electromagnetic waves transmitted and reflected in specific frequency bands can be manipulated. They are currently widely used to design various functional devices, such as frequency-selective surfaces, perfect absorbers, and amplitude modulators. However, with the increasing diversity and complexity of metasurface applications, the design optimization of metasurface unit cell structures has become more challenging. On the one hand, the design objectives of metasurface unit cell structures not only need to meet specific electromagnetic parameters but also other requirements of the metasurface device application scenarios, such as small unit cell size and limited fabrication area. On the other hand, structural optimization cannot destroy the geometric and topological characteristics of the original unit cell structure. This confines the structural parameters within a complex constraint space, greatly increasing the difficulty of the optimization process. Therefore, research on multi-objective optimization methods for metasurface unit cell structures that can satisfy complex constraints is indispensable.

[0003] Existing multi-objective optimization methods for metasurface unit structures fall into two categories: those based on numerical simulation software and those based on neural networks. Methods based on numerical simulation software require simulating the electromagnetic response of the metasurface unit and then calculating the fitness of the optimization algorithm based on this simulation. These methods require extensive numerical simulations, especially when complex inequality constraints need to be satisfied. They consume significant computational resources and time searching within the parameter space that does not meet the constraints, resulting in low search efficiency and slow convergence with small population sizes. Methods based on neural networks require pre-creating large-scale datasets to train the neural network to replace the numerical simulation software before applying it to structure optimization. These methods also suffer from the problem of blindly searching within the parameter space that does not meet the inequality constraints. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention proposes an electromagnetic metasurface unit structure optimization method based on constraint space mapping. By using direction vectors and scaling factors, multidimensional structural variables are mapped to a convex constraint space composed of inequality constraints. Combined with a multi-objective optimization algorithm, this method achieves high-efficiency multi-objective optimization of metasurface unit structures that satisfy complex constraints.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] An optimization method for electromagnetic metasurface unit structures based on constrained space mapping includes the following steps:

[0007] 1) Determine the structural variables to be optimized for the electromagnetic metasurface unit structure, construct the objective function, and determine the inequality constraints on the structural variables;

[0008] 2) Establish the mapping relationship from structural variables to the convex constraint space composed of inequality constraints;

[0009] 3) Initialize the basic parameters of the genetic algorithm according to the objective function and mapping relationship, and construct genetic individuals. Each individual includes a genetic vector and a complete vector, and is randomly initialized;

[0010] 4) Based on the structural variables contained in the complete vector of an individual, model and simulate the metasurface unit structure, calculate the objective function value, and then evaluate the fitness of the individual based on the objective function value;

[0011] 5) Calculate the Pareto rank and crowding distance of each individual in the current population, and select the parent individuals to participate in the genetic operation based on the Pareto rank and crowding distance;

[0012] 6) Perform crossover and mutation operations on the genetic vectors of the selected parent individuals to obtain the genetic vectors of the offspring; obtain the complete vectors of the offspring from the genetic vectors of the offspring according to the mapping relationship, thereby obtaining the offspring individuals, and then evaluate the fitness of the offspring individuals through step 4).

[0013] 7) Combine all offspring individuals and all individuals in the original population into a set, select individuals to form the next generation population, and repeat the iteration. After the iteration is completed, extract the structural variable vector from the complete vector of the individuals in the final population to optimize the electromagnetic metasurface unit structure.

[0014] Further, the method for establishing the mapping relationship from the structural variable to the convex constraint space composed of inequality constraints in step 2) is as follows: select a fixed point in the convex constraint space to represent the position of a structural variable, and establish a mapping relationship, which includes a scaling factor, an upper bound of the modulus, and a direction vector; wherein, the modulus vector is calculated according to the relationship between the structural variable and the inequality constraints, and then the minimum value in the modulus vector is calculated to obtain the upper bound of the modulus; according to the upper bound of the modulus, the range of the scaling factor is adjusted to satisfy the inequality constraints.

[0015] Furthermore, the basic parameters initialized in step 3) include: number of objective functions, population size, total number of iterations, number of structural variables to be optimized, crossover coefficient, coefficient of variation, crossover rate, mutation rate, genetic upper bound vector, and genetic lower bound vector.

[0016] Furthermore, in step 3), the genetic vector is a column vector composed of direction vectors and scaling coefficients, and the complete vector is a column vector composed of structural variable vectors.

[0017] Furthermore, step 3) involves the random initialization of the genetic vector and the complete vector, which includes:

[0018] Set an initial fixed point, and randomly generate a direction vector and a scaling factor for each genetic individual to obtain the genetic vector;

[0019] Based on the established mapping relationship, the randomly generated genetic vector is transformed into the corresponding structural variable vector to obtain the complete vector.

[0020] Furthermore, in step 5), the Pareto rank and crowding distance of each individual in the current population are calculated using the fast non-dominated solution sorting algorithm and the crowding distance calculation algorithm.

[0021] Further, in step 5), the parent individuals participating in crossover and mutation are selected based on Pareto rank and crowding distance. The steps include: randomly selecting two individuals from the population for comparison, and selecting the individual with the higher Pareto rank of the complete vector; if the ranks are the same, comparing the crowding distance, and selecting the individual with the larger crowding distance of the complete vector; if the crowding distances are the same, randomly selecting one of them; the sampling of individuals uses sampling with replacement; the operation is repeated until the new population size reaches the original population size.

[0022] Further, in step 7), after forming a set of all offspring individuals and all individuals in the original population, the Pareto rank and crowding distance of the complete vector of the individuals in the set are calculated; individuals are sorted from smallest to largest according to Pareto rank, and for individuals of the same rank, they are sorted from largest to smallest according to crowding distance, and finally only the top N individuals are retained as the next generation population; the parent individuals participating in the genetic operation are selected through step 5), and the chromosome population is evolved.

[0023] The beneficial effects achieved by this invention are as follows:

[0024] 1. This invention maps multidimensional structural variables to a convex constraint space composed of inequality constraints, which can effectively limit the search range of the algorithm and avoid invalid simulation of structures that do not meet the constraints, thereby significantly improving optimization efficiency and accelerating convergence speed.

[0025] 2. This invention simplifies the generation process of effective initial solutions under complex constraints by using direction vectors and scaling factors to represent structural variables, while enhancing the algorithm's global search capability in the constraint space, which helps to achieve excellent optimization results under small population conditions.

[0026] 3. By limiting the search space to the constraint space, this invention can quickly converge to a local optimum under complex constraints, and by combining a multi-objective genetic algorithm, it can effectively find the Pareto optimum, providing an efficient solution for the multi-objective optimization of metasurface unit structures. Attached Figure Description

[0027] Figure 1 This is a schematic diagram of the electromagnetic metasurface unit structure to be optimized.

[0028] Figure 2 This is a flowchart of a multi-objective optimization algorithm based on constraint space mapping.

[0029] Figure 3 This is a graph showing the algorithm's search curve.

[0030] Figure 4 This is the final optimized Pareto front plot.

[0031] Figure 5 This is the optimized metasurface unit structure diagram.

[0032] Figure 6 It is a simulated S21 curve from 22GHz to 28GHz. Detailed Implementation

[0033] To make the various technical features, advantages, or effects of the present invention more apparent and understandable, the following embodiments are described in detail with reference to the accompanying drawings.

[0034] This invention specifically discloses a method for optimizing electromagnetic metasurface unit structures based on constrained space mapping, selecting a design frequency of 22GHz to 28GHz, and the electromagnetic metasurface unit structure to be optimized as follows: Figure 1 As shown, the unit cell is square with a side length D of 1.42 mm. The dielectric constant of dielectric layer 1 is 6.874, and the tangent loss angle is 0.0161°; the dielectric constant of dielectric layer 2 is 2.1°; the dielectric constant of dielectric layer 3 is 6.756, and the tangent loss angle is 0.0133°; the etched layer is approximately a thin ohmic sheet with a sheet resistance of 0.6 Ω. The simulation frequency band is 22 GHz to 28 GHz, and the number of sampling points is 1001. The processing steps of this method are as follows:

[0035] Step 1. Establish a mathematical model for multi-objective optimization of metasurface unit structures.

[0036] The structural parameters of the metasurface unit structure to be optimized are: There are a total of 6 dimensional parameters, i.e., n=6.

[0037] Construct the minimum objective function minF i ,i∈{1,2,...,k}, where Fi Let k be the i-th objective function, and k be the number of objective functions. In this embodiment, there are 3 objective functions, i.e., k = 3, which are:

[0038]

[0039]

[0040]

[0041] These three objective functions correspond to the average transmission loss, longitudinal etching ratio, and etching area ratio from 24.25 GHz to 25.25 GHz, respectively.

[0042] Based on the modeling expression of structural variables, the inequality constraints are expressed as follows: Where A is the coefficient matrix, with m rows corresponding to the number of inequality constraints, and n columns corresponding to the number of structural variables; The constraint vector has 1 column and m rows, corresponding to the constants in the inequality constraints.

[0043] In this embodiment, the specific constraints are as follows: There are 12 in total, i.e., m = 12.

[0044] The corresponding A equals b equals

[0045] 2. Establish the mapping relationship from multidimensional structural variables to the convex constraint space composed of inequality constraints.

[0046] Any structure variable vector in a convex constraint space consisting of inequality constraints It is possible to obtain a fixed point located in the convex constrained space. Indicate: Where β is the proportionality coefficient, β∈[0,1], a is the upper bound of the modulus, and α≥0. Let be the direction vector, with dimension n. The formula for calculating the upper bound α of the magnitude is: in Let m be the magnitude vector, and m be the dimension. The meaning is the magnitude vector. The j-th dimension. Magnitude vector. The calculation formula is in Let be the constraint vector for the inequality constraints. If When one dimension is 0, or When one dimension is less than 0, The corresponding dimension value is set to +∞. When there exists... When one dimension of the expression is equal to 0, it needs to be checked. If the minimum value among all dimensions other than 0 satisfies the constraint, then the upper bound of the modulus α is equal to the minimum value; otherwise, the upper bound of the modulus α is equal to 0.

[0047] 3. Electromagnetic metasurface unit structures based on constrained space mapping, such as... Figure 2 The process is shown.

[0048] 3.1 Initialize basic algorithm parameters. The number of objective functions is k = 3, the population size is N = 40, the total number of iterations is Gen = 150, the current iteration number is iter, the number of structural variables to be optimized is n = 6, the crossover coefficient is δ1 = 20, the mutation coefficient is δ2 = 20, the crossover rate is 1, the mutation rate is γ = 1 / 7, and the upper bound vector of genetic individuals is... The lower bound vector of genetic individuals is

[0049] 3.2 Constructing the individual genetic vector and the complete vector of an individual. The genetic vector refers to the vector that participates in crossover and mutation operations, while the complete vector refers to the vector that participates in fitness calculation and is also used as the optimization result. The genetic vector is represented as follows: It is composed of direction vectors A column vector consisting of the scaling factor β. The complete vector is represented as... It is composed of structural variable vectors The column vectors formed by these vectors. The genetic vector and complete vector of the t-th individual in the population are represented as follows: and Where t∈{1,2,...,N}. There is a one-to-one correspondence between the genetic vector and the complete vector.

[0050] 3.3 Randomly generate the initial genetic vector and the complete vector. First, set the initial fixed points. Then the genetic vectors of all individuals Set as in Let β be the direction vector corresponding to the genetic vector of the t-th individual in the population. t rand is the proportionality coefficient corresponding to the genetic vector of the t-th individual in the population. n×1 rand is a column vector of dimension n whose elements are uniformly distributed between 0 and 1. 1×1 To obtain uniformly distributed random numbers between 0 and 1, the mapping relationship from step 2 is used to calculate... corresponding

[0051] 3.4 Assess individual fitness. According to... The system incorporates structural variables, performs corresponding metasurface unit modeling and simulation to obtain the required electromagnetic parameters, and calculates the average transmission loss from 24.25 GHz to 25.25 GHz based on these parameters. The longitudinal etching ratio and etching area ratio are calculated based on the structural vector. Finally, the fitness of the complete individual vector is evaluated based on the calculated objective function values; the individual fitness is equal to the vector composed of the three objective function values.

[0052] 3.5 Using the fast non-dominated solution sorting algorithm and the crowding distance calculation algorithm, the Pareto rank and crowding distance of each complete individual vector in the current population are obtained.

[0053] 3.6 Based on Pareto rank and crowding distance, the genetic individual vectors participating in crossover and mutation are selected using the binary tournament principle. Specifically, two individuals are randomly selected from the population for comparison; the individual with the higher Pareto rank is chosen. If the ranks are the same, the crowding distance is compared, and the individual with the larger crowding distance is chosen. If the crowding distances are the same, one is randomly selected. Sampling with replacement is used for individual selection. This process is repeated until the new population size reaches the original population size.

[0054] 3.7 Simulate binary crossover and polynomial mutation on the obtained genetic individual vectors to obtain the genetic individual vectors of the offspring. Then, according to the mapping relationship established in step 2, transform the obtained offspring genetic individual vectors into corresponding structural variable vectors to obtain the complete offspring individual vectors. Finally, evaluate the fitness of the complete individual vectors according to step 3.4.

[0055] 3.8 Based on Pareto rank and crowding distance, an elite retention strategy is used to generate a new generation of the population. Specifically, the complete individual vectors of all offspring obtained in step 3.7 and the original complete individual vectors are combined into a set. The Pareto rank and crowding distance of this set are calculated according to step 3.5. Individuals are sorted in ascending order of Pareto rank. For individuals with the same rank, they are sorted in descending order of crowding distance. Finally, only the top N chromosomes are retained as the next generation of chromosomes. Then, iter is incremented by 1, and the process returns to step 3.6 to iteratively evolve the chromosome population until iter equals Gen, at which point the iteration stops, and the optimization is complete.

[0056] Hypervolume (HV) is used to measure the overall performance of the optimization algorithm, such as... Figure 3 As shown. The algorithm converges after 50 generations, and the final optimized Pareto front is as follows. Figure 4 As shown. To achieve a trade-off among the three objectives, the point [0.502 0.448 0.107 0.0821 0.0584 0.0136] in the Pareto front is chosen. TAs the final optimization result, the metasurface unit structure generated from the optimized structural variables is as follows: Figure 5 As shown, its simulation S21 results from 22GHz to 28GHz are as follows: Figure 6 As shown, the optimized structural parameters enable the metasurface to achieve an average transmission loss of 3.88 dB in the 5G NR n251 band, a longitudinal etching ratio of 89.1%, and an etching area of ​​35.3%.

[0057] Although the present invention has been disclosed above with reference to embodiments, it is not intended to limit the present invention. Appropriate modifications or equivalent substitutions made by those skilled in the art to the technical solutions of the present invention should be covered within the protection scope of the present invention, which is defined by the claims.

Claims

1. A method for optimizing the structure of electromagnetic metasurface units based on constrained space mapping, characterized in that, Includes the following steps: 1) Determine the structural variables to be optimized for the electromagnetic metasurface unit structure, construct the objective function, and determine the inequality constraints for the structural variables; the inequality constraints are expressed as follows: ,in Let be the coefficient matrix, with the following number of rows: The number of columns corresponds to the number of inequality constraints. This corresponds to the number of structure variables; The vector is a restricted vector with 1 column and 1 row. , which corresponds to the constant in the inequality constraint; 2) Establish a mapping relationship from structural variables to the convex constraint space formed by inequality constraints. Specifically, this means that any structural variable vector within the convex constraint space formed by inequality constraints... , by a fixed point located in the convex constrained space Indicate: ,in This is the proportionality coefficient. , The upper bound of the model length, , Let be the direction vector, with dimension . Upper bound of the modulus The calculation formula is ,in Let be a vector of magnitude, with dimension . , The meaning is the magnitude vector. The Dimension; Magnitude vector The calculation formula is ,in Let be the constraint vector of the inequality constraint; if When one dimension is 0, or When one dimension is less than 0, The value of the corresponding dimension is set to When exists When one dimension of is equal to 0, test Does the minimum value among all dimensions other than 0 satisfy the constraints? If so, what is the upper bound of the modulus? It equals the minimum value; if it does not satisfy the condition, the upper bound of the modulus is... Equal to 0; 3) Initialize the basic parameters of the genetic algorithm according to the objective function and mapping relationship, and construct genetic individuals. Each individual includes a genetic vector and a complete vector, and initializes them randomly. 4) Based on the structural variables contained in the complete vector of an individual, model and simulate the metasurface unit structure, calculate the objective function value, and then evaluate the fitness of the individual based on the objective function value; 5) Calculate the Pareto rank and crowding distance of each individual in the current population, and select the parent individuals to participate in the genetic operation based on the Pareto rank and crowding distance; 6) Perform crossover and mutation operations on the genetic vectors of the selected parent individuals to obtain the genetic vectors of the offspring; obtain the complete vectors of the offspring from the genetic vectors of the offspring according to the mapping relationship, thereby obtaining the offspring individuals, and then evaluate the fitness of the offspring individuals through step 4). 7) Combine all offspring individuals with all individuals in the original population into a set, select individuals to form the next generation population, and repeat the iteration process. After iteration, the structural variable vector is extracted from the complete vector of the individuals in the final population to optimize the electromagnetic metasurface unit structure.

2. The method as described in claim 1, characterized in that, The basic parameters initialized in step 3) include: number of objective functions, population size, total number of iterations, number of structural variables to be optimized, crossover coefficient, coefficient of variation, crossover rate, mutation rate, genetic upper bound vector, and genetic lower bound vector.

3. The method as described in claim 1, characterized in that, In step 3), the genetic vector is a column vector composed of direction vectors and scaling coefficients, and the complete vector is a column vector composed of structural variable vectors.

4. The method as described in claim 1, characterized in that, Step 3) involves the random initialization of the genetic vector and the complete vector, which includes: Set an initial fixed point, and randomly generate a direction vector and a scaling factor for each genetic individual to obtain the genetic vector; Based on the established mapping relationship, the randomly generated genetic vector is transformed into the corresponding structural variable vector to obtain the complete vector.

5. The method as described in claim 1, characterized in that, In step 5), the Pareto rank and crowding distance of each individual in the current population are calculated using the fast non-dominated sorting algorithm and the crowding distance calculation algorithm.

6. The method as described in claim 1, characterized in that, In step 5), parent individuals participating in crossover and mutation are selected based on Pareto rank and crowding distance. The steps include: randomly selecting two individuals from the population for comparison, and selecting the individual with the higher Pareto rank of the complete vector; if the ranks are the same, comparing the crowding distance, and selecting the individual with the larger crowding distance of the complete vector; if the crowding distances are the same, randomly selecting one of them; the sampling of individuals uses sampling with replacement; the operation is repeated until the new population size reaches the original population size.

7. The method as described in claim 1, characterized in that, In step 7), after forming a set of all offspring individuals and all individuals in the original population, the Pareto rank and crowding distance of the complete vector of the individuals in the set are calculated. Individuals are sorted from smallest to largest according to their Pareto rank, and for individuals of the same rank, they are sorted from largest to smallest according to their crowding distance. Finally, only the top N individuals are retained as the next generation population. The parent individuals participating in the genetic operation are selected through step 5) to evolve the chromosome population.

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