Multi-objective optimization method for silicon micro resonant accelerometer based on improved NSGA-II

Abstract

This invention discloses a multi-objective optimization method for silicon microresonant accelerometers based on an improved NSGA-II. First, a structural size optimization model for the silicon microresonant accelerometer is established. Then, the improved NSGA-II is used to solve the multi-objective optimization problem, which includes at least the following steps: initialization of the optimal point set population, initial population screening, fast non-dominated sorting, crowding distance calculation, selection, adaptive crossover and Cauchy mutation, and local search. After several iterations, the algorithm obtains a solution set that approximates the true Pareto front. By balancing the performance indicators and robustness of the accelerometer structure, the decision-maker can select a solution as the optimization scheme as needed, and substitute this solution into the accelerometer structure for finite element simulation verification. This invention takes into account the working mode frequency of the mass block, the scaling factor, and the fundamental frequency of the resonator in the silicon microresonant accelerometer, achieving an optimal trade-off between the working mode frequency and the scaling factor. The improvements to NSGA-II expand the population distribution range and make the solution set distribution more uniform.

Description

Technical Field

[0001] This invention belongs to the technical field of microelectromechanical systems (MEMS) and microinertial systems, and relates to the structural design method of silicon microresonant accelerometers, specifically to a multi-objective optimization method for silicon microresonant accelerometers based on an improved NSGA-II. Background Technology

[0002] In practical research on silicon microresonant accelerometers, the accelerometer's fundamental frequency, scaling factor, and resonator fundamental frequency are crucial performance indicators. However, these indicators often present contradictions; improving one performance indicator may lead to the deterioration of another or several others. Previous studies have largely focused on optimizing these performance indicators through qualitative analysis of certain structural parameters, followed by repeated adjustments to these parameters to achieve a satisfactory target performance. This approach is tedious and fails to ensure adequate coordination among the performance indicators, thus failing to achieve optimal compromises.

[0003] NSGA-II is a classic algorithm widely used in the field of multi-objective optimization, but it also has some limitations in practical applications, such as uneven population distribution, getting trapped in local optima, difficulty in maintaining population diversity, and weak local search ability. Summary of the Invention

[0004] This invention addresses the shortcomings of existing technologies, such as the difficulty in fully coordinating various performance indicators of accelerometers and the problems of NSGA-II being prone to getting trapped in local optima, difficulty in maintaining population diversity, and weak local search capabilities. It provides a multi-objective optimization method for silicon microresonant accelerometers based on an improved NSGA-II. First, a structural size optimization model for the silicon microresonant accelerometer is established. Then, the improved NSGA-II is used to solve a multi-objective optimization problem, which includes at least the following steps: initialization of the optimal point set population, initial population screening, fast non-dominated sorting, crowding calculation, selection, adaptive crossover and Cauchy mutation, and local search. After several iterations, the algorithm obtains a solution set that approximates the true Pareto front. By balancing the performance indicators and robustness of the accelerometer structure, decision-makers can select a solution as the optimization scheme as needed, and then substitute this solution into the accelerometer structure for finite element simulation verification. This invention takes into account the working mode frequency of the mass block, the scaling factor, and the fundamental frequency of the resonator of the silicon micro-resonant accelerometer, and achieves optimal compromise by fully coordinating the working mode frequency of the mass block and the scaling factor; the improvements made to NSGA-II expand the population distribution range and make the solution set distribution more uniform.

[0005] To achieve the above objectives, the technical solution adopted by this invention is: a multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II, comprising at least the following steps:

[0006] S1: Establish a multi-objective optimization model for the structural dimensions of a silicon microresonant accelerometer. The model includes an objective function, constraints, and decision variables. The objective function of the model is:

[0007] F1 = max{the working modal frequency of the mass block f M}

[0008] F2 = max{scale factor SF}

[0009] The constraints of the model are:

[0010] f 01 ≤Resonator fundamental frequency f0≤f 02

[0011] Where f 01 f is the lower bound of the resonator's fundamental frequency target range. 02 This represents the upper limit of the resonator's fundamental frequency target range.

[0012] The decision variables of the model are all the structural geometric parameters of the silicon micro-resonant accelerometer;

[0013] S2: The silicon micro-resonant accelerometer structure is optimized based on the improved NSGA-II. The optimization aims to improve the objective function value of the model described in S1. Under the constraints, the Pareto front is obtained through steps such as initialization of the optimal point set population, initial screening, fast non-dominated sorting, crowding distance calculation, selection, adaptive crossover, Cauchy mutation and local search.

[0014] S3: By analyzing the Pareto front obtained from step S2, and weighing the performance indicators and robustness of the accelerometer structure, the decision-maker can select the optimal solution as the final optimization scheme according to the requirements, and substitute the solution into the accelerometer structure dimensions for finite element simulation verification. If the verification results do not meet the design requirements, then repeat steps S2 and S3; if the verification results meet the design requirements, then the multi-objective optimization of the dimensions is completed.

[0015] As an improvement of the present invention, in the objective function of the model in step S1, the working mode frequency f of the mass block is... M The specific methods for obtaining it are as follows:

[0016] S11: Calculate the stiffness k of the accelerometer support beam along the sensitive axis. s :

[0017]

[0018] Where E is the Young's modulus of silicon, w s l is the width of the supporting beam. s t is the length of the supporting beam and t is the thickness of the structural layer;

[0019] S12: Calculate the axial output stiffness k of the lever vo :

[0020]

[0021] Where l is the length of the resonant beam, w is the width of the resonant beam, and l o w is the length of the lever output beam. o The lever outputs the beam width;

[0022] S13: Calculate the input stiffness of the accelerometer lever:

[0023]

[0024] Among them, L o L is the length of the lever's output lever arm. i The input lever arm length;

[0025] S14: Mass block operating mode frequency f M for:

[0026]

[0027] Where M is the mass of the mass block.

[0028] As an improvement of the present invention, the scaling factor SF in the objective function of the model in step S1 is obtained in the following specific way:

[0029] S11': Calculate the axial stiffness k of the lever fulcrum beam vp The rotational stiffness k of the lever fulcrum beam sp and the rotational stiffness k of the lever output end so :

[0030]

[0031] Among them, w p For the width of the supported beam, l p The length of the supported beam;

[0032] S12': Calculate the lever theoretical magnification factor A L :

[0033]

[0034] Among them, L o This refers to the length of the lever's output lever arm.

[0035] S13': Calculate the amplification factor A0 of the lever system on the input inertial force.

[0036]

[0037] Among them, A L This is the leverage factor based on the lever theory.

[0038] S14': The scaling factor SF is:

[0039]

[0040] Where S is the effective area of ​​the mass block, S c ρ represents the total area of ​​the comb tooth region, and ρ represents the silicon density.

[0041] As another improvement of the present invention, the calculation process of the resonator fundamental frequency f0 in the constraint conditions of the model in step S1 is as follows:

[0042] S11”: Calculate the lateral stiffness K of the resonant beam:

[0043]

[0044] Where E is the Young's modulus of silicon, t is the thickness of the structural layer, l is the length of the resonant beam, and w is the width of the resonant beam; S12”: Calculate the equivalent mass M of the resonant beam. eq :

[0045] M eq =0.396ρlwt+ρS c t;

[0046] Where, S c ρ is the total area of ​​the comb tooth region, and ρ is the silicon density.

[0047] S13”: The resonator's fundamental frequency f0 is specifically:

[0048]

[0049] As another improvement of the present invention, step S2 specifically includes the following steps:

[0050] S21: Set population parameters and genetic operation parameters, including but not limited to population size, crossover probability, variable crossover length, maximum number of iterations, and search range parameters;

[0051] S22: Based on the population parameters determined in step S21, initialize the population using the optimal point set method and calculate the objective function value of each individual in the initial population.

[0052] S23: Remove individuals that do not meet the constraints or whose decision variables exceed the range of values ​​during initialization from the population to complete the initial screening;

[0053] S24: Sort the individuals in the population according to the non-dominance level and calculate the crowding distance for each individual;

[0054] The method for determining the level of non-dominance of an individual is as follows:

[0055] For each individual, record two values: the number of individuals that dominate it, n. i (Initially 0) and the set of individuals it governs, S i ;

[0056] Compare the dominance relationships of all individuals pairwise. If the individual Dominant Individual but n i Add 1, and join in S i ;

[0057] Let all n i =0 individuals have a non-dominant rank of 1;

[0058] For each individual with Rank=1, decrease its S i n of the individuals i When an individual's n i When the value is reduced to 0, its non-dominant rank is set to k+1;

[0059] Repeat until the Rank value of all individuals is determined;

[0060] The method for calculating congestion distance is as follows:

[0061] Assuming there are M objective functions and n individuals at a given dominance level, the individuals are sorted in descending order according to their objective function values. The first and last individuals in the sorted list are called boundary points, and their crowding distance on the m-th objective is set to +∞.

[0062]

[0063] For a non-boundary point, its crowding distance on the m-th target is:

[0064]

[0065] in, and These are the minimum and maximum values ​​of the entire population on the m-th target.

[0066] An individual's crowding distance is the sum of their crowding distances to each target:

[0067]

[0068] Repeat until the crowding distance of all individuals of the dominance level is obtained;

[0069] S25: Based on the determined population size, individuals with higher dominance levels are preferentially selected to form a new population, and individuals with the same dominance level are preferentially selected based on the greater crowding distance.

[0070] S26: Using the new population selected in step S25 as the parent population, perform adaptive crossover and Cauchy mutation operations, calculate the objective function value of the offspring population generated by crossover and mutation, and merge the offspring population with the parent population.

[0071] S27: Perform non-dominated sorting and crowding distance calculation again on the population merged in step S26;

[0072] S28: Use the boundary points and sparse points in the population as search centers to perform local search operations, and merge the individuals obtained from the search operations with the population to form a new population.

[0073] S29: Perform fast non-dominated sorting and crowding calculation on the population obtained in step S28;

[0074] S210: Select individuals to form a new population based on dominance order and crowding degree, wherein the number of individuals in the new population is equal to the population size;

[0075] S211: Repeat steps S23 to S210 until the algorithm reaches the maximum number of iterations, at which point the iteration stops and the Pareto front solution set is obtained.

[0076] As another improvement of the present invention, step S22, which initializes the population using the optimal point set method, specifically includes the following steps:

[0077] S221: Calculate the set of optimal points in Euclidean geometric space

[0078]

[0079] Where r is the set of best points in Euclidean geometric space, r j Let p be the set of best points in the j-th spatial dimension, p be the smallest prime number satisfying p≥2s+3, and s be the spatial dimension of the input.

[0080] S222: Constructing the optimal point set:

[0081] P s (i)={(r1i1,r2i2,…,r s i s )}, i=1,2,…,s;

[0082] S223: Set the best points P s Mapped to the feasible region where the population resides:

[0083]

[0084] Among them, a j Let b represent the lower bound of the j-th dimension. j This represents the upper bound of the j-th dimension.

[0085] As another improvement of the present invention, in step S26, the crossover operator of the adaptive crossover operation is calculated as follows:

[0086]

[0087] Where I t I is the current iteration number. t,max P represents the maximum number of iterations. c,min P is the minimum crossover probability. c,max This represents the maximum crossover probability.

[0088] The Cauchy mutation formula in the Cauchy mutation operation is as follows:

[0089]

[0090] Where Cauchy(0,1) is the standard Cauchy distribution, and rand is a random number that follows a uniform distribution in [0,1]. Let m represent the j-th dimension element of the i-th mutated individual. uj Let be the variable length of the j-th dimension element.

[0091] As a further improvement of the present invention, the search mutation formula used in the local search operation of step S28 is:

[0092] x' k =x k +rand(γ)(u k -l k ), 1≤k≤9

[0093] Where rand(γ) represents a random number between (-γ, γ); u k and l k These represent the upper and lower limits of the value of the k-th decision variable, respectively.

[0094] Compared with the prior art, the technical advantages and effects of this invention are as follows:

[0095] (1) This invention introduces a multi-objective optimization method into the structural size optimization of silicon micro-resonant accelerometers, which can simultaneously take into account multiple accelerometer design parameters. Compared with the structural size optimization method that involves multiple adjustments to the structural size and finite element simulation, this invention can greatly shorten the time for accelerometer structural size optimization.

[0096] (2) The optimization method of the present invention uses the optimal point set method to initialize the population, so that the initial population is more evenly distributed in the interval, which is conducive to expanding the optimization range.

[0097] (3) The method of the present invention adopts an adaptive crossover operation, which makes the population have a small crossover probability when the dispersion is high in the early stage and a large crossover probability when the population converges to the optimal Pareto front in the later stage, thus preventing the population from getting trapped in local optima; the method adopts Cauchy mutation operation to make the mutation range more uniform, improve the algorithm's search ability, and avoid the algorithm from getting trapped in local optima to a certain extent.

[0098] (4) The method of the present invention adopts local search operation. Searching with boundary points as the center can expand the population distribution range, and searching with sparse points as the center can increase the uniformity of population distribution. Attached Figure Description

[0099] Figure 1 This is a flowchart illustrating the steps of the multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II of this invention.

[0100] Figure 2 This is a flowchart illustrating the steps of the improved NSGA-II algorithm in the method of this invention.

[0101] Figure 3 The Pareto front is obtained in the example. Detailed Implementation

[0102] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0103] Example 1

[0104] A multi-objective optimization method for silicon microresonant accelerometer structures based on an improved NSGA-II algorithm, such as... Figure 1 As shown, it includes the following steps:

[0105] Step S1: Based on the design requirements of the silicon micro-resonant accelerometer, determine the objective function, constraints, decision variables and their value ranges, and establish a multi-objective optimization model for the silicon micro-resonant accelerometer structure;

[0106] The objective function is:

[0107] F1 = max{the working modal frequency of the mass block f M}

[0108] F2 = max{scale factor SF}

[0109] The working mode frequency f of the mass block M The calculation process is as follows:

[0110] Calculate the stiffness k of the supporting beam along the sensitive axis. s :

[0111]

[0112] Where E is the Young's modulus of silicon, w s l is the width of the supporting beam. s t is the length of the supporting beam and t is the thickness of the structural layer;

[0113] Calculate the axial output stiffness k of the lever vo :

[0114]

[0115] Where l is the length of the resonant beam, w is the width of the resonant beam, and l o w is the length of the lever output beam. o The lever outputs the beam width;

[0116] Calculate the input stiffness k of the accelerometer lever L :

[0117]

[0118] Among them, L o L is the length of the lever's output lever arm. i The input lever arm length;

[0119] Then the working modal frequency f of the mass block M for:

[0120]

[0121] Where M is the mass of the mass block;

[0122] The scaling factor SF is calculated as follows:

[0123] Calculate the axial stiffness k of the lever fulcrum beam vp and rotational stiffness k sp :

[0124]

[0125] Among them, w p For the width of the supported beam, l p The length of the supported beam;

[0126] Calculate the rotational stiffness k at the output end of the lever so :

[0127]

[0128] Calculate the leverage ratio A according to the theory of leverage L :

[0129]

[0130] Calculate the amplification factor A0 of the lever system for the input inertial force:

[0131]

[0132] Among them, A L This is the leverage factor based on the lever theory.

[0133] The scaling factor SF is:

[0134]

[0135] Where S is the effective area of ​​the mass block, S c ρ is the total area of ​​the comb tooth region, and ρ is the silicon density.

[0136] The constraints are as follows:

[0137] f 01 ≤Resonator fundamental frequency f0≤f 02

[0138] Where f 01 f is the lower bound of the resonator's fundamental frequency target range. 02 This represents the upper limit of the resonator's fundamental frequency target range.

[0139] The calculation process for the fundamental frequency f0 of the resonator is as follows:

[0140] Calculate the lateral stiffness K of the resonant beam:

[0141]

[0142] Calculate the equivalent mass M of the resonant beam eq :

[0143] M eq =0.396ρlwt+ρS c t;

[0144] Then the fundamental frequency f0 of the resonator is:

[0145]

[0146] The decision variables include: the width w of the support beam. s Support beam length l s L, the length of the lever output arm o L, the length of the lever input arm i The width of the resonant beam is w, the length of the resonant beam is l, and the width of the lever output beam is w. o Lever output beam length lo The width of the lever fulcrum beam w p Length l of the lever fulcrum beam p .

[0147] In this embodiment, ρ = 2330 kg / m 3 E = 1.7 × 10 11 t = 80 μm, M = 4.36 μg. The decision variables and their ranges are shown in Table 1.

[0148] Table 1 Decision Variables and Their Ranges

[0149]

[0150]

[0151] Step S2: Optimize the structural dimensions of the silicon microresonant accelerometer using the improved NSGA-II; such as... Figure 2 As shown, the specific steps include the following:

[0152] S21. Set population parameters and genetic operation parameters: Determine the population size, crossover probability, variable crossover length, maximum number of iterations, and search range parameters;

[0153] The population size N is 100, and the minimum crossover probability P is... c,min The crossover probability is 0.5, and the maximum value P is... c,max The maximum number of iterations is 1. t,max The value is 200, the search range parameter γ is 1, and the variable asynchronous length is m. uj =0.1(u j -l j ), u j and l j These are the upper and lower bounds of the range of values ​​for the j-th decision variable, respectively.

[0154] S22. Population initialization: Based on the determined population size and the range of decision variables, the optimal point set method is used to initialize the population, and the objective function value of each individual in the initial population is calculated.

[0155] S23. Initial screening: Remove individuals that do not meet the constraints from the population;

[0156] S24. Fast non-dominated sorting and crowding calculation: Sort individuals in the population according to their non-dominated levels and calculate the crowding of each individual.

[0157] S25. Selection: Based on the determined population size, individuals with higher dominance levels are selected to form a new population. For individuals with the same dominance level, those with greater crowding distance are selected.

[0158] S26, Crossover and Mutation: Perform adaptive crossover and Cauchy mutation operations on the population generated in the previous step as the parent population, calculate the objective function value of the offspring population generated by crossover and mutation, and merge the offspring population with the parent population.

[0159] S27. Non-dominated sorting and crowding distance calculation: Perform non-dominated sorting and crowding distance calculation on the population obtained in the previous step.

[0160] S28. Local search: Use the boundary points and sparse points in the population as the search center to perform a local search operation, and merge the individuals obtained by the search operation with the population to form a new population.

[0161] S29. Fast non-dominated sorting and crowding calculation: Perform fast non-dominated sorting and crowding calculation on the population obtained in the previous step.

[0162] S210. Selection: Based on the order of dominance and crowding, individuals are selected to form a new population, and the number of individuals in the new population is equal to the population size.

[0163] S211. Repeat steps S23 to S210 until the algorithm reaches the maximum number of iterations.

[0164] Step S3: Plot the individuals with the highest dominance level (Pareto front) in the population obtained from the last iteration of Step S2 as a scatter plot, as shown below. Figure 3 As shown, by balancing the performance indicators and robustness of the accelerometer structure, a solution is selected as the final optimized scheme. This solution is then substituted into the accelerometer structure dimensions for finite element simulation verification to ensure the optimization effect. If the verification result does not meet the design requirements, steps S2 and S3 are repeated. In summary, the method of this invention takes into account the working mode frequency of the mass block, the scaling factor, and the fundamental frequency of the resonator in the multi-objective optimization process of the silicon microresonator accelerometer structure dimensions. It uses an improved NSGA-II to solve this multi-objective optimization problem, selects a suitable solution from the obtained Pareto front as the final optimized scheme, and substitutes this solution into the accelerometer structure dimensions for finite element simulation verification. This method fully coordinates the working mode frequency of the mass block and the scaling factor to achieve an optimal trade-off, greatly shortening the optimization time of the accelerometer structure dimensions and improving the algorithm performance.

[0165] It should be noted that the above content merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. For those skilled in the art, various improvements and modifications can be made without departing from the principle of the present invention, and all such improvements and modifications fall within the scope of protection of the claims of the present invention.

Claims

1. A multi-objective optimization method for silicon micro resonant accelerometer based on improved NSGA-II, characterized in that, It should include at least the following steps: S1: Establish a multi-objective optimization model for the structural dimensions of a silicon microresonant accelerometer. The model includes at least an objective function, constraints, and decision variables. The objective function of the model is: = max{Mass block operating mode frequency }; = max{scale factor }; The constraints of the model are: ≤Resonator fundamental frequency ≤ ; in This represents the lower bound of the resonator's fundamental frequency target range. This represents the upper limit of the resonator's fundamental frequency target range. The decision variables of the model are all geometric parameters of the silicon micro-resonant accelerometer structure; In the objective function of the model, the working modal frequency of the mass block is... The specific methods for obtaining it are as follows: S11: Calculate the stiffness of the accelerometer support beam along the sensitive axis. : ； in, The Young's modulus of silicon. The width of the supporting beam, The length of the supporting beam, The thickness of the structural layer; S12: Calculate the axial output stiffness of the lever : ； in, The length of the resonant beam. The width of the resonant beam. The length of the lever output beam. The lever outputs the beam width; S13: Calculate the input stiffness of the accelerometer lever: ； in, The length of the lever's output lever arm. The input lever arm length; S14: Operating mode frequency of the mass block for: ； in, For the mass of the mass block; In the objective function of the model, the scaling factor The specific methods for obtaining it are as follows: S11': Calculate the axial stiffness of the lever fulcrum beam Lever fulcrum beam rotational stiffness and the rotational stiffness of the lever output end : ； in, The width of the support beam, The length of the supported beam; S12': Calculate the theoretical leverage ratio : ； in, This refers to the length of the lever's output lever arm. S13': Calculate the amplification factor of the lever system on the input inertial force. : ； in, This is the leverage factor based on the lever theory. S14': Scale factor for: ； in The effective area of ​​the mass block. This represents the total area of ​​the comb teeth region. Silicon density; S2: The structural dimensions of the silicon micro-resonant accelerometer are optimized based on the improved NSGA-II. The optimization process aims to improve the objective function value of the model described in step S1. Under the constraints, the process involves at least the following steps: initialization of the optimal point set population, initial screening, fast non-dominated sorting, crowding distance calculation, selection, adaptive crossover, Cauchy mutation, and local search, to obtain the Pareto front. The local search involves using the boundary points and sparse points in the population as search centers for local search operations. S3: Using the Pareto front obtained in step S2, the decision-maker selects the optimal solution as the final optimization scheme according to the requirements, and substitutes the solution into the accelerometer structure dimensions for finite element simulation verification. If the verification result does not meet the design requirements, steps S2 and S3 are repeated; if the verification result meets the design requirements, the multi-objective optimization of the dimensions is completed.

2. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 1, characterized in that: In the constraints of the model in step S1, the fundamental frequency of the resonator... The calculation process is as follows: S11'': Calculate the lateral stiffness of the resonant beam : ； in, The Young's modulus of silicon. For the thickness of the structural layer, The length of the resonant beam. The width of the resonant beam; S12'': Calculate the equivalent mass of the resonant beam : ； in, This represents the total area of ​​the comb teeth region. Silicon density; S13'': Resonator fundamental frequency Specifically: 。 3. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 1, characterized in that: Step S2 specifically includes the following steps: S21: Set population parameters and genetic operation parameters, including but not limited to population size, crossover probability, variable crossover length, maximum number of iterations, and search range parameters; S22: Based on the population parameters determined in step S21, initialize the population using the optimal point set method and calculate the objective function value of each individual in the initial population. S23: Remove individuals that do not meet the constraints or whose decision variables exceed the range of values ​​during initialization from the population to complete the initial screening; S24: Sort the individuals in the population according to the non-dominance level and calculate the crowding distance for each individual; S25: Based on the determined population size, individuals with higher dominance levels are preferentially selected to form a new population, and individuals with the same dominance level are preferentially selected based on the greater crowding distance. S26: Using the new population selected in step S25 as the parent population, perform adaptive crossover and Cauchy mutation operations, calculate the objective function value of the offspring population generated by crossover and mutation, and merge the offspring population with the parent population. S27: Perform non-dominated sorting and crowding distance calculation again on the population merged in step S26; S28: Use the boundary points and sparse points in the population as search centers to perform local search operations, and merge the individuals obtained from the search operations with the population to form a new population. S29: Perform fast non-dominated sorting and crowding distance calculation on the population obtained in step S28; S210: Select individuals to form a new population based on dominance order and crowding degree, wherein the number of individuals in the new population is equal to the population size; S211: Repeat steps S23 to S210 until the algorithm reaches the maximum number of iterations, at which point the iteration stops and the Pareto front solution set is obtained.

4. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 3, characterized in that: In step S22, initializing the population using the optimal point set method specifically includes the following steps: S221: Calculate the set of optimal points in Euclidean geometric space ； in, It is the set of best points in Euclidean geometric space. For the first A set of ideal points in a spatial dimension To meet The smallest prime number, The spatial dimension of the input quantity; S222: Constructing the optimal point set: ； S223: Collection of Best Points Mapped to the feasible region where the population resides: ； in, Indicates the first The lower bound of the dimension Indicates the first The upper limit of dimensions.

5. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 3 or 4, characterized in that: In step S26, the crossover operator of the adaptive crossover operation The calculation method is as follows: ； in This represents the current iteration number. The maximum number of iterations, This represents the minimum crossover probability. This represents the maximum crossover probability. The Cauchy mutation formula in the Cauchy mutation operation is as follows: ； in It is a standard Cauchy distribution. For random numbers that follow a uniform distribution in [0,1], Indicates the first The first mutant individual dimensional elements, For the first The variable length of a dimension element.

6. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 5, characterized in that: In the local search operation of step S28, the search mutation formula used is: ； in Indicates that the value is in (- , A random number between ) and They represent the first The upper and lower limits of the values ​​that a decision variable can take.

7. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 3, characterized in that: The non-dominated sorting in step S2 specifically involves recording two values ​​for each individual: the number of individuals that dominate it. and the set of individuals it governs ; Compare the dominance relationships of all individuals pairwise. If the individual Dominant Individual ,but of Add 1, and join in of ; Make all =0 individuals have a non-dominant rank of 1; For each individual with Rank=1, reduce its Individual When an individual's When the rank is reduced to 0, set its non-dominant rank to k+1; Repeat until the Rank value of all individuals is determined.

8. The multi-objective optimization method for silicon micro-resonant accelerometers based on the improved NSGA-II as described in claim 7, characterized in that: The method for calculating the congestion distance in step S2 is as follows: Assuming there is a total There are several objective functions, and the number of individuals at a certain dominance level is... The calculation sorts the individuals in descending order according to their objective function values. The first and last individuals in the sorted list are called boundary points, and their values ​​are... The crowding distance on each target is set as follows: : ； For non-boundary points, their position on the 1st... The crowding distance on each target is: ； in, and Is the entire population in the first Minimum and maximum values ​​for each objective; An individual's crowding distance is the sum of their crowding distances to each target: 。

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