Gradient-based acoustic metamaterial continuum topological optimization method

Abstract

According to the gradient-based acoustic metamaterial continuum topological optimization method provided by the invention, a structure with excellent sound absorption performance under a target frequency is obtained by optimizing the distribution of a rigid material in a porous material layer. The method comprises the following steps: defining a finite element model, dividing grids and setting related parameters; design variables are introduced, and design domain material attributes are determined through a unified multiphase modeling (UMP) technology, namely a material interpolation function; deriving an acoustic equation weak form calculation unit mass matrix, a stiffness matrix and a load matrix; assembling the matrix, and solving required acoustic parameters; establishing a topological optimization model; solving the sensitivity of the constraint function through an adjoint method; the MMA optimization algorithm is combined with density filtering and projection filtering to continuously update design variables, and finally an excellent material topology layout is obtained, so that the sound absorption coefficient at the target frequency is close to 1.

Description

A gradient-based method for topology optimization of acoustic metamaterial continuum Technical Field

[0001] This invention relates to the field of acoustic finite element topology optimization technology, specifically to a gradient-based acoustic metamaterial continuum topology optimization method. Background Technology

[0002] Noise can severely damage human hearing and accelerate the aging of buildings and mechanical structures. Noise pollution is considered a serious environmental problem worldwide, and noise and vibration control has always been a hot research topic. Porous materials are widely used in sound-absorbing materials due to their excellent sound dissipation capabilities. When homogeneous porous materials are installed on acoustic hard walls, the absorption efficiency at low frequencies is severely limited unless the porous material layer is thick enough, as the first peak frequency of the sound absorption curve corresponds to a quarter-wavelength resonance. By embedding horizontal and vertical rigid baffles in the porous layer, the multi-thickness resonance can be effectively extended by achieving surface impedance matching, thus improving low-frequency sound absorption performance. However, currently, this method is mainly adopted from a forward design perspective, relying on experience and inspiration, which is often a very difficult task. Furthermore, considering multiple target frequencies during design not only incurs high computational costs but also carries a high probability of failure. Topology optimization based on numerical methods can effectively solve this problem.

[0003] Topology optimization is a computer-aided design method that, under given constraints, uses mathematical algorithms to find the optimal distribution of materials to achieve a specific objective. Currently, variable density methods, level set methods, and genetic algorithms are widely used in topology optimization. The variable density method discretizes the design domain into a finite element mesh, assigning a continuous density variable to each element to describe the material distribution within the domain. Considering all factors, the variable density method can effectively solve topology optimization problems involving the distribution of rigid materials within porous material layers.

[0004] The invention disclosed in CN120068531A discloses a method for optimizing the layout of porous material layers for noise reduction on the surface of acoustic cavities. The method includes: 1) constructing a material interpolation model for the noise-reducing porous material layer using a variable density method; 2) establishing a topology optimization model with a sound absorption coefficient close to 1 at the target frequency as a constraint function; 3) calculating the sensitivity of the constraint function using the adjoint method; and 4) iteratively solving the topology optimization mathematical model using a moving asymptote algorithm to obtain the optimal layout of the noise-reducing porous material layer on the acoustic cavity surface. However, this sound-absorbing topology optimization method is currently only applicable to two-phase materials; further exploration is needed for topology optimization of multiphase materials. Summary of the Invention

[0005] The purpose of this invention is to provide a gradient-based topology optimization method for acoustic metamaterial continuums. This method can optimize rigid materials within porous materials to obtain a good sound absorption coefficient at a target frequency. Furthermore, the concept of this method can be extended to the optimization of other acoustic metamaterials, addressing the problem of insufficient optimization design capabilities for acoustic components.

[0006] To achieve the above-mentioned technical objectives, the technical solution adopted by the present invention is as follows:

[0007] A gradient-based method for topology optimization of acoustic metamaterial continuums, the method comprising the following steps:

[0008] S1, Define the finite element model, determine the optimized and non-optimized regions and corresponding boundary conditions, divide the mesh and set the basic acoustic parameters;

[0009] S2 introduces a design variable with values ​​of 0 and 1 to distinguish between porous and rigid materials. By using a unified multiphase modeling technique, the equivalent bulk modulus and equivalent density of porous and rigid materials in the optimization region are determined.

[0010] S3, construct the weak form of the Helmholtz equation, and calculate the mass matrix, stiffness matrix, and load matrix of the finite element;

[0011] S4, based on the mass matrix, stiffness matrix and load matrix, assembles the overall solution linear matrix and calculates the acoustic parameters of the component;

[0012] S5. A topology optimization model is established with the sound absorption coefficient at the target frequency being close to 1 as a constraint.

[0013] S6, calculate the sensitivity of the objective function and constraint function of the topology optimization model to the independent variables;

[0014] S7. The sensitivity of the objective function and constraint function to the independent variables obtained in step S6 is used as the key input parameters of the MMA optimization algorithm. The MMA optimization method is applied to update the independent variables. Then, density filtering is performed on the design variables to avoid the checkerboard effect. Finally, the Heaviside projection function is used to perform projection filtering on the design variables to update the design variables.

[0015] S8, determine all design variables If the convergence is 0 or 1, proceed to step S3 to continue iterating; if it is, terminate the iteration. All materials eventually converge to rigid or porous materials, and the structure represented by the topology optimization model that meets the conditions is obtained.

[0016] Step S1 further includes:

[0017] In the programming tool, the optimized and non-optimized regions are defined, a structured second-order quadrilateral mesh is used, and the number and size of elements in each geometric direction are set to divide the acoustic finite element mesh.

[0018] The basic acoustic parameters are set as interpolation model penalty factor, air density and bulk modulus, atmospheric sound pressure, sound velocity and sensitivity filter radius.

[0019] Step S2 further includes:

[0020] Introducing continuous design variables e = 1, 2, ..., N, where N represents the total number of finite element elements in the design domain. When the design variables... When =1, it indicates that the material property of the finite element element is a porous material, and when the design variables... =0 indicates that the material property of the finite element is a rigid material;

[0021] By using a unified multiphase modeling technique, the equivalent bulk modulus and equivalent density of porous and rigid materials in the optimization region are characterized as design variables. Interpolation function:

[0022] ;

[0023] ;

[0024] in and The density and bulk modulus of the porous material are obtained using the JCA parameters. and The density and bulk modulus of the rigid material are obtained through a unified multiphase model.

[0025] Step S3 further includes:

[0026] The weak form of the Helmholtz equation is constructed as follows:

[0027] ;

[0028] Among the symbols For weighted sound pressure, Represented as the particle displacement vector in the acoustic medium; and Represents the density and bulk modulus of the medium in the acoustic region, and the regional sound pressure is... angular frequency is , Represents the Laplace operator;

[0029] The stiffness matrix of the finite element is calculated using the following formula. quality matrix and load matrix :

[0030] ,

[0031] ,

[0032] ;

[0033] Among the symbols and The expression represents the density and bulk modulus of the acoustic medium of the corresponding unit. and The components of particle displacement along the x-axis and y-axis in the acoustic medium. and The components of the boundary external normal vector along the x-axis and y-axis.

[0034] Step S4 further includes:

[0035] Based on the mass matrix, stiffness matrix, and load matrix, the linear matrix for assembling the overall solution is as follows:

[0036] ;

[0037] By using a linear matrix, the sound pressure at each node of each finite element element is obtained, and the sound pressure is also obtained from the design variables. Determined sound absorption coefficient and reflection coefficient :

[0038] ;

[0039] In the formula, This is represented as taking the design variables. The absolute value of the reflection coefficient;

[0040] ;

[0041] In the formula, and Positions and The sound pressure at that location; This represents the wavenumber corresponding to the non-optimized region.

[0042] Further, in step S5, the topology optimization model is:

[0043]

[0044] St.

[0045] ;

[0046] In the formula, Let be the objective function. For constraint functions, To determine the number of design variables, Representing different constraint target frequencies, , This is expressed as the number of target frequencies. Indicates tolerance.

[0047] Further, in step S6, the sensitivity of the objective function to the independent variable is calculated using the following formula:

[0048] ;

[0049] The adjoint method is used to obtain the sensitivity of the constraint function to the independent variable. Specifically, the Lagrange multipliers are calculated through the adjoint equation. The sensitivity of the constraint function is then expressed as follows:

[0050] ;

[0051] In the formula, For the system matrix, and Each is determined by design variables The determined stiffness matrix and mass matrix For state variables, is a Lagrange multiplier vector.

[0052] Furthermore, in step S7, the MMA optimization method is applied to construct an optimization model to update the independent variables. The key steps of the general MMA optimization algorithm are given below, namely, constructing the convex subproblem:

[0053]

[0054] For this topology optimization model (the meaning of the parameters of the optimization model is detailed in claim 6), in the first... The next iteration, given the current point For constraint functions Construct the following convex and separable approximation function (the objective function in this paper). (For convex functions, no approximation function needs to be constructed):

[0055]

[0056] in and The coefficients of the upper asymptote and the lower asymptote, respectively, are given by the following formula:

[0057] ,

[0058] ;

[0059] In the formula, For the first In the nth iteration The values ​​of the variables; and The first The problem is thus transformed into a convex and separable problem using an upward and downward shifting asymptote of each variable. We solve it using a highly efficient and robust dual convex optimization algorithm.

[0060] Furthermore, the design variables are density filtered using the following density filtering function:

[0061] ;

[0062] In the formula, Represented as a weight function, , For the filter radius, The distance between two design variables. For units within the filtration radius, For the selected design variables;

[0063] Use the following Havisend projection function to perform projection filtering on the design variables:

[0064] ;

[0065] In the formula, The parameters that determine the steepness of the projection function are increased with the number of iterations to help the element material converge.

[0066] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0067] This invention presents a gradient-based topology optimization method for acoustic metamaterial continuums. It proposes a topology optimization model for rigid material distribution within porous materials and utilizes the adjoint matrix method to solve for constraint function sensitivity in the sound absorption problem, significantly improving the solution speed. Ultimately, a satisfactory sound-absorbing structure is obtained, providing a reference for subsequent acoustic metamaterial topology optimization. Attached Figure Description

[0068] Figure 1 is a schematic diagram of the gradient-based acoustic metamaterial continuum topology optimization method of the present invention; Figure 2 is a schematic diagram of the finite element model; Figure 3 is a schematic diagram of the iteration curve of the objective function; Figure 4 is a schematic diagram of the iteration curve of the constraint function and the topology under different iteration steps; Figure 5 is a comparison diagram of the sound absorption coefficient of the final structure and the sound absorption coefficient of pure porous material. Detailed Implementation

[0069] The embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0070] Referring to Figure 1, this invention discloses a gradient-based topology optimization method for acoustic metamaterial continuums, the method comprising the following steps:

[0071] S1, Define the finite element model, determine the optimized and non-optimized regions (air domain in this implementation) and the corresponding boundary conditions, divide the mesh and set the basic acoustic parameters;

[0072] S2 introduces a design variable with values ​​of 0 and 1 to distinguish between porous and rigid materials. By using the Unified Multiphase Modeling (UMP) technique, the equivalent bulk modulus and equivalent density of porous and rigid materials in the optimization region are determined.

[0073] S3, construct the weak form of the Helmholtz equation, and calculate the mass matrix, stiffness matrix, and load matrix of the finite element;

[0074] S4, based on the mass matrix, stiffness matrix and load matrix, assembles the overall solution linear matrix and calculates the acoustic parameters of the component;

[0075] S5. A topology optimization model is established with the sound absorption coefficient at the target frequency being close to 1 as a constraint.

[0076] S6, calculate the sensitivity of the objective function and constraint function of the topology optimization model to the independent variables;

[0077] S7. The sensitivity of the objective function and constraint function to the independent variables obtained in step S6 is used as the key input parameters of the MMA optimization algorithm. The MMA optimization method is applied to update the independent variables. Then, density filtering is performed on the design variables to avoid the checkerboard effect. Finally, the Heaviside projection function is used to perform projection filtering on the design variables to update the design variables.

[0078] S8, determine all design variables If the convergence is 0 or 1, proceed to step S3 to continue iterating; if it is, terminate the iteration. All materials eventually converge to rigid or porous materials, and the structure represented by the topology optimization model that meets the conditions is obtained.

[0079] The optimization method of the present invention will be illustrated below through an example.

[0080] S1: Define the optimized and non-optimized regions (see attached figures). Use a structured second-order quadrilateral mesh, set the number of elements and element size in each geometric direction, and divide the acoustic finite element mesh.

[0081] As shown in Figure 2, the left side of the finite element model is defined as the incident boundary condition, and the other three sides are defined as hard acoustic wall boundary conditions. The gray area represents the design region, and x1 and x2 are the sensor positions. A quadrilateral structured mesh is used to mesh the air domain and the design region. The basic acoustic parameters are set as the interpolation model penalty factor, air density and bulk modulus, atmospheric sound pressure, sound velocity, and sensitivity filter radius.

[0082] S2: Introducing continuous design variables (e=1,2,…N; N: total number of elements in the design domain), where design variables The upper limit is 1, and the lower limit is 0. When When =1, the material properties of the element are represented as those of a basic porous material. When the density is 0, it represents a rigid material. By using a unified multiphase modeling technique and determining the equivalent bulk modulus and equivalent density of the two materials through interpolation, the density, a property characterizing the material, is established. and bulk modulus For the following design variables Interpolation function:

[0083]

[0084]

[0085] in , The density and bulk modulus of the porous material are obtained using JCA parameters. , The density and bulk modulus of the rigid material are obtained through a unified multiphase model.

[0086] In this example, the equivalent density and equivalent bulk modulus of porous materials are determined using the JCA parameters. The density and bulk modulus of the following rigid materials are determined using a unified multiphase technique:

[0087] , ;

[0088] The equivalent density and equivalent bulk modulus of the material are then determined by an interpolation function with the design variables as independent variables.

[0089] S3: The governing equation for the linear steady-state acoustic problem is the Helmholtz equation:

[0090] ;

[0091] in and These represent the density and bulk modulus of the medium in the acoustic region. Note that this varies depending on the type of medium in the acoustic region. and Taking different values, the regional sound pressure level is angular frequency is , Represents the Laplace operator.

[0092] The specific derivation of the weak form of the Helmholtz equation is as follows:

[0093] ;

[0094] Among the symbols For weighted sound pressure, It is represented as the particle displacement vector of the acoustic medium.

[0095] Where the stiffness matrix quality matrix and load matrix The definition is as follows:

[0096] ,

[0097] ,

[0098] ;

[0099] Among the symbols and The expression represents the density and bulk modulus of the acoustic medium of the corresponding unit. and The components of particle displacement along the x-axis and y-axis in the acoustic medium. and The components of the boundary outward normal vector along the x-axis and y-axis are... It represents angular frequency.

[0100] S4: The linear matrix for assembling the overall solution is as follows:

[0101] ;

[0102] Using this linear matrix, the sound pressure level at each node of each unit can be calculated. The sound absorption coefficient can then be determined. and reflection coefficient This prepares for solving the constraint function and its sensitivity later. The sound absorption coefficient, a key parameter in acoustics describing the sound absorption capacity of a material or structure, is defined as the ratio of the sound energy absorbed by the material to the incident sound energy, ranging from 0 to 1. It is calculated using the following formula:

[0103] ;

[0104] Reflectance coefficient Defined as the complex ratio of the sound pressure of the reflected wave to the sound pressure of the incident wave. This is represented as taking the design variables. The absolute value of the reflection coefficient is determined. For this example, The numerical value can be obtained by analogy to the method of measuring the sound absorption coefficient in an impedance tube and is expressed as the following formula:

[0105] ;

[0106] in , For position and The sound pressure at that location; This is expressed as the wavenumber corresponding to the air domain.

[0107] S5: Research shows that the sound absorption coefficient at the target frequency... Using a maximum value of 1 as a constraint in the equations ultimately yields good results. The final topology optimization model is established as follows:

[0108]

[0109] St.

[0110]

[0111] In the formula, This is a tolerance factor used to help the constraint function converge; the tolerance can be considered to be 0.05-0.5 depending on the specific problem. This invention considers the multi-objective case, with the frequencies of the two objectives set as... =1100Hz, =2100Hz. Consider using The mathematical form used as the objective function can help the unit material converge to porous or rigid materials, where This represents the number of design variables. The objective function reaches its minimum value of 0 when each unit material converges to either 0 or 1. Equation 16 represents the constraint equation, which ensures that the sound absorption coefficient is close to 1 at the target frequency, while also considering tolerance settings. It helps with convergence. Representing different constraint target frequencies , This represents the number of target frequencies. It can be noted that when pursuing sound absorption coefficients at multiple target frequencies, setting the objective function in this way ensures good convergence performance of the material, resulting in a clear layout and configuration of the two materials.

[0112] S6: To solve the aforementioned optimization problem, gradient-based optimization methods require calculating the sensitivity of the objective function and constraint equations to the design variables. The sensitivity of the objective function can be directly obtained as follows:

[0113] ;

[0114] The sensitivity of the constraint equations is obtained using the adjoint method, which has a computational complexity of O(n log n). The direct differentiation method and the adjoint method have a computational complexity of only [missing information]. Its calculation cost and the number of design variables This significantly reduces computation time. Verification shows that the sensitivity of the constraint equations obtained through the adjoint method is consistent with the results obtained through the direct differentiation method, as detailed below:

[0115] First, construct the following Lagrange variables:

[0116] ;

[0117] in For the system matrix, For state variables, is a Lagrange multiplier vector.

[0118] The derivative of the design variable can be written as follows:

[0119]

[0120] After rearranging, we get the following formula:

[0121]

[0122] because The arbitrariness of the value allows for the selection of an appropriate one. make The term coefficient is zero, thus avoiding expensive solutions. .

[0123] Find the appropriate solution The adjoint equation for the value can be expressed as follows:

[0124] ;

[0125] Note that since the state variables are complex variables and the constraint functions... If the constraint function is a real-valued function, then in the optimization problem, the gradient direction of the constraint function is determined by the constraint function with respect to the state variables. conjugate The decision is made, and the value is... The above formula can be rewritten as follows:

[0126] ;

[0127] Using Wirtinger's derivative rule, the right-hand side of the above equation can be written as follows:

[0128] ;

[0129] The subscripts r and i denote the real and imaginary parts of the complex number, respectively. The Lagrange multipliers are calculated using the adjoint equation. Then, the sensitivity of the constraint function can be expressed as follows:

[0130] ;

[0131] The derivative of the load term with respect to the design variables naturally disappears.

[0132] S7: Moving Asymptote (MMA) is a gradient-based optimization method that prioritizes satisfying constraint equations when updating design variables, making it well-suited for this problem. The key steps of this optimization algorithm in this problem are given below:

[0133]

[0134] For this topology optimization model, in the... The next iteration, given the current point For constraint functions Construct the following convex and separable approximation function (the objective function in this paper). (For convex functions, no approximation function needs to be constructed):

[0135]

[0136] in and The coefficients of the upper asymptote and the lower asymptote, respectively, are given by the following formula:

[0137] ,

[0138] ;

[0139] In the formula, For the first In the nth iteration The values ​​of the variables; and The first The problem is thus transformed into a convex and separable problem using an upward and downward shifting asymptote of each variable. We solve it using a highly efficient and robust dual convex optimization algorithm.

[0140] The density filtering function is as follows:

[0141] ;

[0142] The weight function is defined as follows:

[0143] ;

[0144] in The filter radius is set to 1.5 times the grid size in this paper. The distance between two design variables.

[0145] The Havisend projection function is as follows:

[0146] ;

[0147] The parameters that determine the steepness of the projection function increase with the number of iterations, helping the element material converge.

[0148] Figure 3 shows the material convergence of the structure at iteration steps 60, 90, 120, and 180.

[0149] Figure 4 shows how the values ​​of the two constraint functions change with the iteration step. Constraint function 1 corresponds to the target frequency of 1100Hz and constraint function 2 corresponds to the target frequency of 2100Hz.

[0150] Figure 5 shows a comparison between the sound absorption coefficient of the final structure and that of the pure porous material. The results indicate that a structure with excellent sound absorption coefficient was successfully obtained at the target frequency, improving the structure's sound absorption performance at low frequencies.

[0151] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0152] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A gradient-based method for topology optimization of acoustic metamaterial continuums, characterized in that, The method includes the following steps: S1, defining a finite element model, determining the optimized and non-optimized regions and corresponding boundary conditions, meshing, and setting basic acoustic parameters; S2, introducing design variables with values ​​of 0 and 1 to distinguish between porous and rigid materials. The steps are as follows: S3, using unified multiphase modeling technology, the equivalent bulk modulus and equivalent density of porous and rigid materials in the optimization region are determined; S4, a weak form of the Helmholtz equation is constructed, and the mass matrix, stiffness matrix, and load matrix of the finite element elements are calculated; S5, based on the mass matrix, stiffness matrix, and load matrix, the linear matrix of the overall solution is assembled, and the acoustic parameters of the component are calculated; S6, with the sound absorption coefficient close to 1 at the target frequency as a constraint, a topology optimization model is established; S7, the sensitivity of the objective function and constraint function of the topology optimization model to the independent variables is calculated; S8, the sensitivity of the objective function and constraint function to the independent variables obtained in step S6 is used as the key input parameters of the MMA optimization algorithm, the MMA optimization method is applied to update the independent variables, density filtering is then applied to the design variables to avoid the checkerboard effect, and finally, the Heaviside projection function is used to project and filter the design variables to update the design variables; S9, all design variables are judged. If the convergence is 0 or 1, proceed to step S3 to continue iterating; if it is, terminate the iteration. All materials eventually converge to rigid or porous materials, and the structure represented by the topology optimization model that meets the conditions is obtained.

2. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 1, characterized in that, Step S1 further includes: defining the optimized and non-optimized regions in the programming tool, using a structured second-order quadrilateral mesh, setting the number of elements and element size in each geometric direction, and dividing the acoustic finite element mesh; setting the basic acoustic parameters as the interpolation model penalty factor, air density and bulk modulus, atmospheric sound pressure, sound velocity, and sensitivity filter radius.

3. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 1, characterized in that, Step S2 further includes: introducing continuous design variables. e = 1, 2, ..., N, where N represents the total number of finite element elements in the design domain. When the design variables... When =1, it indicates that the material property of the finite element element is a porous material, and when the design variables... =0 indicates that the material property of the finite element element is a rigid material; by using a unified multiphase modeling technique, the equivalent bulk modulus and equivalent density of porous and rigid materials in the optimization region are characterized as design variables. Interpolation function: ； ;in and The density and bulk modulus of the porous material are obtained using the JCA parameters. and The density and bulk modulus of the rigid material are obtained through a unified multiphase model.

4. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 1, characterized in that, Step S3 further includes: constructing the weak form of the Helmholtz equation as follows: ; among which symbols For weighted sound pressure, Represented as the particle displacement vector in the acoustic medium; and Represents the density and bulk modulus of the medium in the acoustic region, and the regional sound pressure is... angular frequency is ， Representing the Laplace operator; the stiffness matrix of the finite element is calculated using the following formula. quality matrix and load matrix ： ， ， ; among which symbols and The expression represents the density and bulk modulus of the acoustic medium of the corresponding unit. and The components of particle displacement along the x-axis and y-axis in the acoustic medium. and The components of the boundary external normal vector along the x-axis and y-axis.

5. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 1, characterized in that, Step S4 further includes: assembling the overall solution linear matrix based on the mass matrix, stiffness matrix, and load matrix as follows: By using a linear matrix, the sound pressure at each node of each finite element element is obtained, and the sound pressure is also obtained from the design variables. Determined sound absorption coefficient and reflection coefficient ： In the formula, This is represented as taking the design variables. The absolute value of the reflection coefficient; In the formula, and Positions and The sound pressure at that location; This represents the wave number corresponding to the non-optimized region, where j is the imaginary unit.

6. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 1, characterized in that, In step S5, the topology optimization model is: ;St. ； In the formula, Let be the objective function. For constraint functions, To determine the number of design variables, Representing different constraint target frequencies, ， This is expressed as the number of target frequencies. Indicates tolerance.

7. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 6, characterized in that, In step S6, the sensitivity of the objective function to the independent variable is calculated using the following formula: The adjoint method is used to obtain the sensitivity of the constraint function to the independent variable. Specifically, the Lagrange multipliers are calculated through the adjoint equation. The sensitivity of the constraint function is then expressed as follows: In the formula, For the system matrix, and Each is determined by design variables The determined stiffness matrix and mass matrix For state variables, is a Lagrange multiplier vector.

8. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 6, characterized in that, In step S7, the MMA optimization algorithm is applied to construct the following optimization model to update the independent variables: ； For this topology optimization model, in the... The next iteration, given the current point For constraint functions Construct the following convex and separable approximate function: ;in and The coefficients of the upper asymptote and the lower asymptote, respectively, are given by the following formula: ， In the formula, For the first In the nth iteration The values ​​of the variables; and The first The upward and downward asymptotes of each variable are used to transform the optimization problem into a convex and separable problem, which is then solved using a dual convex optimization algorithm.

9. The gradient-based acoustic metamaterial continuum topology optimization method according to claim 1, characterized in that, In step S7, the following density filtering function is used to filter the design variables: In the formula, Represented as a weight function, ， For the filter radius, The distance between two design variables. For units within the filtration radius, For the selected design variables, use the following Havisend projection function to perform projection filtering on the design variables: In the formula, The parameters that determine the steepness of the projection function are increased with the number of iterations to help the element material converge.

Images