Non-gradient and gradient collaborative topology optimization method of single-phase elastic superstructure

Abstract

This invention discloses a non-gradient and gradient-coordinated topology optimization method for a single-phase elastic superstructure, comprising: generating a random initial configuration and encoding the random initial configuration and design variables into real numbers, denoted as X; constructing a multi-objective optimization model based on X that considers maximizing bandgap performance and maximizing structural connectivity, to convert the wave vector... k The constraints are limited to characteristic paths on the boundaries of irreducible Brillouin zones. A non-dominated sorting genetic algorithm is used to solve these constraints to obtain the Pareto front. High-performance candidate solutions are selected from the Pareto front as initial inputs. Based on the high-performance candidate solutions, a topology optimization model considering bandgap performance maximization and structural connectivity as constraints is constructed to solve for the current design variables. A three-field floating projection topology optimization framework is introduced to update the design field design variables. Implicit floating projection constraints are then introduced to further iterate and update the design field design variables until convergence.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of elastic superstructure, and particularly relates to a non-gradient and gradient collaborative topology optimization method of single-phase elastic superstructure. BACKGROUND

[0002] Functional elastic superstructure has wide application prospects in the fields of sound wave and elastic wave propagation control, vibration isolation and energy directional transmission due to its unique wave modulation characteristics. Such materials can be regarded as a kind of heterogeneous medium with artificial microstructure, and their physical properties mainly come from the topological configuration and material distribution of the internal periodic structure, rather than simply from the material intrinsic properties. Through active design of the lattice structure, unconventional behaviors such as bandgap effect, negative parameter characteristics and directional wave propagation can be achieved in single-phase or multi-phase systems.

[0003] The bandgap characteristic of elastic superstructure is one of its most remarkable physical characteristics, that is, it prohibits the propagation of sound waves or vibrations within a specific frequency range. This phenomenon is due to the Bragg scattering and local resonance effect at the interface of the periodic structure, and the bandgap width and position are mainly affected by the cell geometry, characteristic size and material parameters. Due to the strong heterogeneity of the superstructure on the wavelength scale, in addition to the bandgap effect, it can also achieve wave energy localization, wave collimation, turning and defect state conduction functions.

[0004] In the design aspect, the topology optimization method can break through the limitations of traditional empirical design by systematically regulating the material distribution in the design domain, and can achieve precise regulation of the physical properties of the superstructure. This method has been widely used in the fields of bandgap widening, frequency response optimization, mechanical property strengthening and multi-physical field coupling performance design of phononic crystals and superstructures. With the development of computational mechanics and optimization algorithms, topology optimization is evolving towards multi-objective, robustness and reliability, providing a theoretical basis and technical support for multi-scale and multi-field collaborative design.

[0005] At present, the topology optimization of phononic crystals and superstructures mainly includes the isotropic solid material penalty method (SIMP) based on gradient information, the floating projection method (FPTO), the level set method, and the genetic algorithm (GA), the simulated annealing algorithm (SA) based on non-gradient information. Gradient-based methods have the advantages of high optimization efficiency and good convergence, but they are prone to local optimal solutions and are strongly dependent on the initial configuration. Although non-gradient algorithms have strong global search ability and high solution diversity, they have high computational cost and strong parameter sensitivity. In addition, in order to meet the manufacturability requirements, an equivalent permeability constraint is often introduced in the design of single-phase superstructure to maintain the connectivity of the structure in the optimization iteration process, but there is a potential competition between this constraint and the bandgap characteristic, making it difficult to simultaneously achieve good permeability and excellent wave regulation performance.

[0006] In summary, existing single-gradient or non-gradient optimization methods have limitations in the design of elastic superstructures: the former struggles to escape local optima, while the latter suffers from low optimization efficiency. Effectively integrating the advantages of both types of algorithms, balancing global search capabilities with local fine-grained optimization capabilities, has become an important research direction in the topology optimization design of elastic superstructures. Therefore, a hybrid topology optimization method is urgently needed to achieve efficient exploration of the design space and collaborative optimization of multi-objective performance, thereby improving the designability and feasibility of elastic superstructures in engineering applications. Summary of the Invention

[0007] In view of this, the present invention provides a non-gradient and gradient-coordinated topology optimization method for single-phase elastic superstructures, which at least solves the problem that existing topology optimization algorithms are difficult to balance global search and local fine optimization.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] A non-gradient and gradient-coordinated topology optimization method for single-phase elastic superstructures includes the following steps:

[0010] Phase 1: Complete NSGA-II multi-objective optimization;

[0011] Generate a random initial configuration for the unit cell, and encode the random initial configuration and design variables using real numbers. The encoded design variable vector is denoted as... ;

[0012] based on A multi-objective optimization model considering both maximizing bandgap performance and maximizing structural connectivity is constructed to transform the wave vector. The constraint is set on the scan path along the boundary of the irreducible Brillouin zone. This constraint is solved using a non-dominated sorting genetic algorithm to obtain... The Pareto optimal solution set converges to the Pareto front after population evolution;

[0013] Phase Two: Complete three FPTO topology optimizations;

[0014] Select high-performance candidate solutions as initial inputs from the Pareto front;

[0015] Based on high-performance candidate solutions, a topology optimization model is constructed that considers the maximization of bandgap performance and structural connectivity as constraints. The current design variables are solved. If the preset convergence condition is not met, a three-field floating projection topology optimization framework is introduced to realize the topology optimization of the single-phase elastic superstructure. The design field design variables are updated, and then implicit floating projection constraints are introduced to further iterate and update the design field design variables until the preset convergence condition is met.

[0016] Preferably, the irreducible Brillouin zone (IBZ) corresponding to the unit cell is the minimal repeating region of the first Brillouin zone after symmetry operations; uniformly distributed wave vector sampling points are set along the boundary of the IBZ, and the scanning path is set as: point M. → Point (0,0) → Point X → point The eigenfrequency corresponding to each wave vector sampling point is obtained, thereby yielding the band dispersion curve. eigenfrequency Follow the wave arrow The changing relationship.

[0017] The preferred multi-objective optimization model considering both maximizing bandgap performance and maximizing structural connectivity is as follows:

[0018] ;

[0019] In the formula, , The design variable vector represents the material distribution state of discrete elements within a unit cell, where 1 represents solid material and 0 represents voids. Indicates the first in the design domain Design variables for discrete units Indicates the total number of discrete units; Let be the first objective function, defined as the... Rank and number The relative band gap width between energy bands, where and They represent the first Rank and first The maximum eigenfrequency of the first energy band, and The first Rank and first The minimum eigenfrequency of the first energy band; The second objective function is defined as the equivalent permeability of the optimized structure. , used to characterize the connectivity of single-phase materials; The equivalent permeability of the discrete unit. It is a two-dimensional or three-dimensional identity matrix. For the design domain volume, It is a trial function, related to two-dimensional or three-dimensional problems; and They are respectively direction and The directional component is a homogenized variable. and They represent direction and The basis vector of direction, It is the unit cell lattice constant. It is a positive real number close to zero.

[0020] Preferably, the problem is solved using a non-dominated sorting genetic algorithm to obtain... The specific contents of the Pareto optimal solution set include:

[0021] (1) Initialize the maximum number of iterations M and the population size. Number of discrete units Randomly generate the initial population Each individual All are 3D design variable vector;

[0022] (2) For the current generation population Finite element analysis was performed on each individual in the population to calculate the objective function value; non-dominated ordination was performed based on Pareto dominance to divide the population into different levels of non-dominated fronts, and crowding distance of each individual was calculated.

[0023] (3) A tournament selection strategy is used to select parent individuals from the current population, and the generation of the second generation is generated by simulating binary crossover (SBX) and polynomial mutation operations. Offspring population ;

[0024] (4) Merging parent populations and offspring population Implement an elite retention strategy, selecting based on non-dominant level and crowding distance. The next generation of the population consists of outstanding individuals. ;

[0025] (5) Repeat (2)-(4) until the maximum iteration number M is reached, and output the final Pareto front solution set, which contains a set of non-dominated optimal solutions that take into account both bandgap performance and structural connectivity.

[0026] Preferably, the specific content of the topology optimization model considering the maximization of bandgap performance and structural connectivity as constraints is as follows:

[0027] ;

[0028] In the formula, The design variables for the second-stage gradient optimization algorithm are represented by the linear difference between the coarse and fine meshes in the first stage, which characterize the material distribution state of the discrete units within the unit cell. 1 represents solid material and 0 represents voids. Let be the objective function of the topology optimization model, representing the th Rank and number The relative band gap width between energy bands, where Indicates the first The maximum eigenfrequency of the first energy band, For the first The minimum eigenfrequency of the first energy band; Equivalent permeability is used to characterize the connectivity of single-phase materials. This is the penetration threshold.

[0029] Preferably, during the FPTO process, dimensionless element design variables are interpolated using a material interpolation model. Mapped to physically meaningful material parameters:

[0030] ;

[0031] In the formula: subscript Indicates solid phase material parameters; For material density, and This represents the Lamé elastic constant.

[0032] Preferably, when the preset convergence condition is not met, a three-field floating projection topology optimization framework is introduced to achieve topology optimization of the single-phase elastic superstructure. The specific content of updating the design field design variables includes:

[0033] If the preset convergence condition is not met, the design field design variables are converted into filter field design variables by density filtering, and then the filter field design variables are mapped to the physical field. The sensitivity of the design field design variables is obtained according to the chain rule, and the design field design variables are iteratively updated according to the sensitivity.

[0034] Preferably, the specific content of converting the design field design variables into filter field design variables through density filtering includes:

[0035] ;

[0036] in, Design variables for the filter field. Design variables and weighting factors for the design field , For the filter radius, For the first The and the first The center distance of each discrete unit Represents a vector matrix of discrete design variables The design variable index in the code.

[0037] Preferably, the specific content of mapping the filter field design variables to the physical field includes:

[0038] ;

[0039] In the formula, For the number of iterations, Indicates the first Physical field design variables in the next iteration process; For projection threshold, The projection sharpness parameter from the filter field to the physical field controls the degree of discretization of the variable. These are the floating projection parameters required for design variable iteration in the design field.

[0040] Preferably, the specific content of obtaining the sensitivity of the design variables of the design field according to the chain rule includes:

[0041] ;

[0042] In the formula, and The objective functions of the topology optimization model are respectively The sensitivity of the constraint function to the design variables of the design field;

[0043] No. The partial derivatives of the first-order intrinsic frequencies with respect to the design variables are:

[0044] ;

[0045] In the formula, the first The partial derivative of the maximum value of the first-order eigenfrequency or the minimum value of the (n+1)th-order eigenfrequency with respect to the design variable is:

[0046] ;

[0047] In the formula, The normalized eigenmode vector; and These are the overall stiffness matrix and the mass matrix, respectively; The intrinsic frequency, For wave vectors;

[0048] ;

[0049] In the formula, the parameter It is a key factor controlling the approximate smoothness of the p-norm;

[0050] ;

[0051] In the formula, Design domain volume; This represents the unit equivalent penetration rate.

[0052] Preferably, the specific content of iteratively updating the design variables of the design field based on sensitivity includes:

[0053] ;

[0054] in, This represents the current iteration number; and They are respectively The objective function of the topology optimization model at -1 iterations The sensitivity of the constraint function to the design variables of the design field; These are Lagrange multipliers used to reinforce the equivalent permeability constraint;

[0055] Furthermore, by introducing implicit floating projection constraints in the design field, the smooth and clear boundary representation is continuously refined and enhanced:

[0056] ;

[0057] Among them, projection threshold Conservation of the sum before and after projection of design variables is achieved through design variable projection. Implicit determination; The floating projection parameters required for design variable iteration in the design field;

[0058] Get Use an incremental approach: starting from the initial value Starting with =1, when the convergence criterion is satisfied... When the precision is less than or equal to the preset precision or the number of predefined iterations is reached, Increasing Δ =1.

[0059] Preferably, when After convergence, the final iterative convergence determination of the optimization algorithm is performed, i.e., the preset convergence condition is:

[0060] Unit solution By interpolating to the pixel scale using level set functions, the smoothed design is then projected back onto the original mesh while preserving the equivalent volume, resulting in a smoothed solution. ;

[0061] The relative performance difference between the smooth solution and the element solution is used to determine the convergence of the second-stage iteration:

[0062] ;

[0063] When the convergence condition is met, the smooth solution is considered equivalent to the element solution, and the corresponding optimized configuration is obtained, thus terminating the optimization.

[0064] Otherwise, continue iterating until the convergence condition is met or the maximum number of iterations is reached.

[0065] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a non-gradient and gradient cooperative topology optimization method for single-phase elastic superstructures, which has the following beneficial effects:

[0066] (1) Achieving the organic unity of global search and local refinement in the topology optimization process.

[0067] By employing a two-stage collaborative optimization strategy of non-gradient and gradient, the global search capability of genetic algorithms and the efficient convergence characteristics of gradient algorithms are effectively integrated. This approach avoids the problem of excessively high computational cost of a single non-gradient algorithm and overcomes the defect of a single gradient algorithm being prone to getting trapped in local optima, thus significantly improving the quality and stability of the optimization results.

[0068] (2) Overcoming the technical bottleneck of collaborative optimization of multiple performance indicators

[0069] An innovative multi-objective optimization framework for equivalent permeability and bandgap width was established, revealing the intrinsic relationship between structural connectivity and wave propagation control performance. This enabled the superstructure to achieve excellent bandgap characteristics while maintaining good manufacturability during the iterative optimization process, laying a theoretical foundation for the engineering application of single-phase superstructures.

[0070] (3) Establish a performance verification system for multi-dimensional wave propagation control

[0071] This invention effectively balances computational cost and optimization quality through a two-stage hybrid strategy, significantly reducing computation time compared to traditional single genetic algorithms. Simultaneously, it significantly improves topology accuracy through real-number encoding and design variable encryption techniques. Furthermore, by obtaining the Pareto optimal solution set, it provides designers with multiple topology options that balance different performance priorities. Gradient optimization using multiple initial solutions significantly enhances the robustness of the optimization results to the initial configuration, effectively improving the diversity of optimization solutions and the stability of the iterative process.

[0072] (4) Establish a performance verification system for multi-dimensional wave propagation control

[0073] This invention not only focuses on traditional bandgap width metrics but also systematically examines the wave manipulation performance of optimized configurations in terms of group velocity distribution within the passband, band anisotropy in different propagation directions, and excitation of defect states / topological states within the bandgap. Through numerical simulation and experimental verification of finite-period array structures, a performance mapping relationship from the infinite-period theoretical model to the actual finite-period structure is established, fully validating the engineering practicality and reliability of this method in diverse wave manipulation applications such as wave propagation attenuation, group velocity modulation, directional propagation, and localized state excitation.

[0074] (5) It has strong universality and is easy to promote and apply.

[0075] The method proposed in this invention is not only applicable to the design of single-phase elastic superstructures, but can also be extended to the topology optimization of other types of superstructures such as acoustic superstructures and electromagnetic superstructures. It has broad engineering application prospects and can be applied to multiple fields such as vibration reduction and noise reduction, waveguide control, and stealth materials. Attached Figure Description

[0076] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0077] Figure 1 A flowchart illustrating a non-gradient and gradient-coordinated topology optimization method for a single-phase elastic superstructure provided by the present invention;

[0078] Figure 2 A non-gradient topology optimization process diagram based on real number encoding and coarse grid (for the third-order bandgap) provided for embodiments of the present invention.

[0079] Figure 3 A gradient topology optimization process diagram based on a high-performance initial solution (for the third-order bandgap) is provided for embodiments of the present invention; wherein: Figure 3 (a) is based on Figure 2 Using the feasible solution a1 from the non-dominated solution set as the initial configuration, under the equivalent permeability constraint... The iterative process of gradient topology optimization under the condition of 0.285 is shown, demonstrating the process of gradually evolving from the initial configuration containing gray units to a clear 0 / 1 topology configuration. Figure 3 (b) is based on Figure 2 Using the feasible solution a2 from the non-dominated solution set as the initial configuration, under the equivalent permeability constraint... The iterative process of gradient topology optimization under the condition of 0.396 is shown, demonstrating the process of gradually evolving from the initial configuration containing gray units to the clear 0 / 1 topology configuration. Figure 3 (c) is based on Figure 2 Using the feasible solution a3 from the non-dominated solution set as the initial configuration, under the equivalent permeability constraint... The iterative process of gradient topology optimization under the condition of 0.612 is shown, demonstrating the process of gradually evolving from the initial configuration containing gray units to a clear 0 / 1 topology configuration.

[0080] Figure 4 The energy band structure based on optimized configuration (for the third-order bandgap) provided in the embodiments of the present invention; wherein: Figure 4 (a) is Figure 3(a) shows the band structure of the final optimized configuration RUC-A1. The horizontal axis represents the irreducible Brillouin zone boundary path (M→Г→X→M), and the vertical axis represents the dimensionless frequency. The shaded area in the figure marks the frequency range of the elastic wave bandgap. Figure 4 (b) is Figure 3 (b) shows the band structure of the final optimized configuration RUC-A2. The horizontal axis represents the irreducible Brillouin zone boundary path (M→Г→X→M), and the vertical axis represents the dimensionless frequency. The shaded area in the figure marks the frequency range of the elastic wave bandgap. Figure 4 (c) is Figure 3 (c) shows the band structure of the final optimized configuration RUC-A3. The horizontal axis represents the irreducible Brillouin zone boundary path (M→Г→X→M), and the vertical axis represents the dimensionless frequency. The shaded area in the figure marks the frequency range of the elastic wave bandgap.

[0081] Figure 5 The present invention provides gradient and non-gradient optimization processes based on higher-order bandgap, and band structures for optimized structures (specifically for sixth-order bandgap); wherein: Figure 5 (a) is an iterative process diagram of non-gradient topology optimization based on real number encoding and coarse grid. A global search is performed on the third-order out-of-plane bandgap, showing the changing trend of the objective function value (bandgap width and equivalent permeability) and the formation process of the Pareto front during the 50 generations of population evolution. Figure 5 (b) is a process diagram of gradient topology optimization based on high-performance initial solution. It shows the complete process of evolution from the initial configuration (containing gray units) obtained by non-gradient optimization to clear 0 / 1 topology driven by floating projection, and finally obtains the optimized configuration RUC-A4. Figure 5 (c) is Figure 5 (b) shows the band structure of the optimized configuration RUC-A4. The shaded area indicates the frequency range of the sixth-order out-of-plane elastic wave bandgap, verifying the effectiveness and robustness of the two-stage hybrid optimization strategy in high-order bandgap design.

[0082] Figure 6 In-plane vibration transmission characteristics of RUC-A2 and A4 finite periodic arrays provided in embodiments of the present invention: numerical and experimental comparison; wherein: Figure 6 (a) shows the vibration transmissivity curves of a 3×3 finite periodic array based on the optimized configuration RUC-A2 under in-plane and out-of-plane harmonic excitation. The horizontal axis is the dimensionless frequency and the vertical axis is the transmissivity (dB). Figure 6 (b) shows the vibration transmissibility curves of a 3×3 finite periodic array based on the optimized configuration RUC-A4 under in-plane and out-of-plane harmonic excitation. The horizontal axis is the dimensionless frequency and the vertical axis is the transmissibility (dB). Detailed Implementation

[0083] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0084] To address the problems of traditional single-gradient topology optimization methods in the design of single-phase elastic superstructures, such as easy getting trapped in local optima, strong dependence on initial configuration, and insufficient diversity of optimization results, as well as the inherent limitations of non-gradient optimization methods, such as high computational cost, low convergence efficiency, and high sensitivity to optimization parameters, and considering the need for coordinated optimization of multiple performance indicators in practical engineering applications of superstructures, especially the conflicting relationship between equivalent permeability (ensuring structural connectivity or manufacturability) and bandgap characteristics, this invention provides a non-gradient and gradient coordinated topology optimization method for single-phase elastic superstructures, comprising the following steps:

[0085] Phase 1: Complete NSGA-II multi-objective optimization;

[0086] Generate a random initial configuration for the unit cell, and encode the random initial configuration and design variables using real numbers. The encoded design variable vector is denoted as... ;

[0087] based on A multi-objective optimization model considering both maximizing bandgap performance and maximizing structural connectivity is constructed to transform the wave vector. The constraint is set on the scan path along the boundary of the irreducible Brillouin zone. This constraint is solved using a non-dominated sorting genetic algorithm to obtain... The Pareto optimal solution set converges to the Pareto front after population evolution;

[0088] Phase Two: Complete three FPTO topology optimizations;

[0089] Select high-performance candidate solutions as initial inputs from the Pareto front;

[0090] Based on high-performance candidate solutions, a topology optimization model is constructed that considers the maximization of bandgap performance and structural connectivity as constraints. The current design variables are solved. If the preset convergence condition is not met, a three-field floating projection topology optimization framework is introduced to realize the topology optimization of the single-phase elastic superstructure. The design field design variables are updated, and then implicit floating projection constraints are introduced to further iterate and update the design field design variables until the preset convergence condition is met.

[0091] It should be noted that:

[0092] The physical properties of elastic superstructures (such as manufacturability, bandgap characteristics, and in-plane stiffness) largely depend on their geometry, material properties, and spatial distribution within the unit cell. To meet the diverse and competing performance requirements in engineering applications, performance-constrained reverse engineering of elastic superstructures is necessary. This invention uses a square unit cell (or other shapes, such as hexagons) as the basic unit, discretizes the material distribution under two-dimensional plane strain conditions, and employs the finite element method to simulate the mechanical superstructure physical properties under different topological configurations, thereby achieving efficient evaluation and optimization of structural performance. With multi-band elastic wave attenuation performance and structural manufacturability as multi-objective optimization directions, this invention establishes a multi-objective topology optimization design framework. It uses specific-order bandgap characteristics as the primary optimization objective and considers structural connectivity (constrained by equivalent permeability) as a secondary objective. Since there is an inherent competitive relationship between bandgap performance and structural connectivity—that is, improving bandgap characteristics often weakens structural connectivity, and vice versa—this optimization framework has significant advantages in achieving performance synergy and objective balance, providing a reliable approach for the efficient design of multifunctional elastic superstructures. Furthermore, the optimization framework of this invention has good versatility and scalability. Its optimization objectives, cell symmetry, and geometry can all be flexibly adjusted and improved according to actual needs to adapt to the performance design requirements of different application scenarios.

[0093] To further implement the above technical solution, the irreducible Brillouin zone (IBZ) corresponding to the unit cell is the minimal repeating region of the first Brillouin zone after symmetry operation simplification; uniformly distributed wave vector sampling points are set along the boundary of the IBZ, and the scanning path is set as: point M. → Point (0,0) → Point X → point The eigenfrequency corresponding to each wave vector sampling point is obtained, thereby obtaining the band dispersion curve. eigenfrequency Follow the wave arrow The changing relationship.

[0094] It should be noted that:

[0095] The primary design objective of topology optimization for single-phase elastic superstructures is to maximize the relative width of the bandgap formed between dispersion curves of a specified order, thereby exploring the design potential for suppressing in-plane vibration waves in the low-frequency broadband range. This invention combines the finite element discretization method with Bloch periodic boundary conditions to solve for dispersion characteristics and elastic wave propagation spectra. Based on the plane strain assumption and linear elastic constitutive relations, the propagation behavior of elastic waves in solid media is described by the following governing equations:

[0096] ;

[0097] In the formula, and Represents Lamé's elastic constant; Characterizing the mass density of a material; and These correspond to the spatial position vector of the material inside the unit cell and the structural response displacement vector, respectively. It is a differential operator. Among them, the particle vibration direction of the inner longitudinal wave (P wave) is in the xy plane and is parallel to the wave vector direction; the particle vibration direction of the in-plane transverse wave (SV wave) is in the xy plane but is orthogonal to the wave vector direction.

[0098] This invention primarily considers the propagation characteristics of elastic waves in the xy-plane. The plane strain assumption efficiently simplifies the three-dimensional model to a two-dimensional form, significantly improving computational efficiency while preserving the thickness-direction constraint effect. It is important to note that the choice of plane stress and plane strain assumptions depends on the specific problem. Therefore, the aforementioned governing equations are decoupled into equations describing in-plane modes.

[0099] ;

[0100] ;

[0101] By applying Bloch periodic boundary conditions to the representative unit cell boundary, the spatiotemporal variables of the displacement field are separated, and its expression is as follows:

[0102] ;

[0103] In the formula, For the displacement field in the spatiotemporal domain, It is a periodic displacement amplitude function. The simplified wave vector within the first Brillouin zone. It is a spatial position vector. Angular frequency, Let i be the time variable, and i be the imaginary unit.

[0104] Substituting the above displacement field expression into the governing equations, and after discretization, we obtain the matrix form of the eigenvalue problem for elastic wave propagation:

[0105] ;

[0106] In the formula, The global stiffness matrix is ​​related to the wave vector. The global mass matrix is ​​independent of the wave vector. This represents the vector of nodal displacement magnitudes. It should be noted that the stiffness matrix... The elements depend on the wave vector. The value of reflects the influence of wave propagation directionality on the structural dynamics, while the mass matrix It is only related to the material distribution and unit cell geometry, and does not change with the wave vector.

[0107] The stiffness matrix expansion and the shape functions of the two-dimensional plane strain four-node isoparametric element for in-plane modes are as follows:

[0108] ;

[0109] ; ;

[0110] ; ;

[0111] ; ;

[0112] ; ; ;

[0113] ; ; ;

[0114] Solving the above eigenvalue problem using the finite element method yields a given wave vector. The eigenfrequency below and the corresponding intrinsic modes, which are This method allows for the construction of complete bandgap dispersion curves, laying the foundation for subsequent bandgap characteristic analysis. respectively wave vector in direction and The directional component is a homogenized variable. It is an imaginary variable. It is a shape function, related to the mesh type. In this embodiment, TPU is used as the material parameter, and the material parameter is set as follows: Young's modulus. =25MPa, Poisson's ratio =0.49, density =1200 kg / m 3 The corresponding shear wave velocity is =45.6 m / s. To standardize the calculation scale and facilitate normalization analysis, the lattice constant is taken as a=1m. In this embodiment, thermoplastic polyurethane elastomer (TPU) is selected as the matrix material for the superstructure design. It has good flexibility and processing adaptability, and is suitable for the fabrication and experimental research of wave-controlled structures.

[0115] Based on representative units The symmetry (fourfold rotational symmetry) corresponds to the irreducible Brillouin zone (IBZ), which is the minimal repeating region after symmetry operations on the first Brillouin zone. For efficient calculation of the band structure, this invention sets uniformly distributed wave vector sampling points along the IBZ boundary, and the scanning path is set to point M. → Point (0,0) → Point X → Point M Thus, the band dispersion curve is obtained. eigenfrequency Follow the wave arrow The changing relationship. When a frequency gap appears between adjacent energy bands, that is, when a certain frequency range exists... When there is no corresponding real wave vector solution, an elastic wave bandgap is formed. Elastic waves within this bandgap frequency range cannot exist in a propagation mode in a periodic structure, and their wave energy decays exponentially, thereby effectively suppressing vibrations in a specific frequency band.

[0116] To further implement the above technical solution, a multi-objective optimization model considering both maximizing bandgap performance and maximizing structural connectivity is specifically proposed as follows:

[0117] ;

[0118] In the formula, , The design variable vector represents the material distribution state of discrete elements within a unit cell, where 1 represents solid material and 0 represents voids. Indicates the first in the design domain Design variables for discrete units Indicates the total number of discrete units; Let be the first objective function, defined as the... Rank and number The relative band gap width between energy bands, where and They represent the first Rank and first The maximum eigenfrequency of the first energy band, and The first Rank and first The minimum eigenfrequency of the first energy band; The second objective function is defined as the equivalent permeability of the optimized structure. , used to characterize the connectivity of single-phase materials; The equivalent permeability of the discrete unit. It is a two-dimensional or three-dimensional identity matrix. For the design domain volume, It is a trial function, related to two-dimensional or three-dimensional problems; and They are respectively direction and The directional component is a homogenized variable. and They represent direction and The basis vector of direction, It is the unit cell lattice constant. It is a positive real number close to zero.

[0119] It should be noted that:

[0120] In this embodiment, the design domain volume Through trial functions =[1,0] T Or [0,1] T To obtain its solution vector, in order to ensure numerical stability and avoid singularities, and The values ​​of are all limited to Within the range.

[0121] Due to the relative band gap width With equivalent penetration rate There is a competitive relationship between the two objectives (i.e., increasing connectivity often leads to a decrease in bandgap performance). This invention employs a non-dominated sorting genetic algorithm (NSGA-II) to solve the aforementioned multi-objective optimization problem, obtaining a Pareto optimal solution set that balances both objectives. The maximum number of iterations is set to M=100, and the population size is set to... =60, grid cell is The population evolves through genetic operators such as selection, crossover, and mutation, ultimately converging to the Pareto front. This optimization model, being dimensionless, is universal and applicable to the topology optimization design of superstructures with different band gaps, material systems, and scales. Its implementation process is visible in [the document / reference needed]. Figure 1 .

[0122] This embodiment uses real-number encoding to represent design variables (encoding level nnum=11), effectively eliminating the precision loss of binary encoding and achieving seamless integration with subsequent gradient optimization. NSGA-II achieves iterative updates of the solution set through a collaborative screening mechanism of non-dominated sorting and crowding distance, effectively preventing premature convergence and expanding the feasible solution space.

[0123] To further implement the above technical solution, a non-dominated sorting genetic algorithm was used to solve it, resulting in... The specific contents of the Pareto optimal solution set include:

[0124] (1) Initialize the maximum number of iterations M and the population size. Number of discrete units Randomly generate the initial population Each individual All are 3D design variable vector;

[0125] (2) For the current generation population Finite element analysis was performed on each individual in the population to calculate the objective function value; non-dominated ordination was performed based on Pareto dominance to divide the population into different levels of non-dominated fronts, and crowding distance of each individual was calculated.

[0126] (3) A tournament selection strategy is used to select parent individuals from the current population, and the generation of the second generation is generated by simulating binary crossover (SBX) and polynomial mutation operations. Offspring population ;

[0127] (4) Merging parent populations and offspring population Implement an elite retention strategy, selecting based on non-dominant level and crowding distance. The next generation of the population consists of outstanding individuals. ;

[0128] (5) Repeat (2)-(4) until the maximum iteration number M is reached, and output the final Pareto front solution set, which contains a set of non-dominated optimal solutions that take into account both bandgap performance and structural connectivity.

[0129] To further implement the above technical solution, the specific content of simulating binary crossover (SBX) and polynomial mutation operations includes:

[0130] ;

[0131] ;

[0132] in, and Let each represent a design variable vector for any two individuals in the population. A random number between [0, 1]; As an intermediate variable; This represents the distribution index of the SBX operator, which controls the similarity between offspring and parents;

[0133] ;

[0134] ;

[0135] in, and These represent the design variables before and after the variation; and Indicates the upper and lower limits of the design variables; This represents the distribution index of the PM operator, which controls the intensity of variation.

[0136] It should be noted that:

[0137] A combination of the simulated binary crossover (SBX) and polynomial mutation (PM) operators is used to balance the global exploration and local exploitation capabilities of the proposed hybrid optimization algorithm in the first optimization stage. In this embodiment, the distribution exponent of the SBX operator... The value ranges from 0 to 5, controlling the similarity between offspring and parents; the distribution index of the PM operator. The value is set to 10-100 to control the mutation intensity. The two operators work together to achieve efficient convergence while ensuring population diversity, thus guaranteeing the acquisition of a high-quality Pareto solution set.

[0138] To further implement the above technical solution, the specific content of the topology optimization model considering bandgap performance maximization and structural connectivity as constraints is as follows:

[0139] ;

[0140] In the formula, The design variables for the second-stage gradient optimization algorithm are represented by the linear difference between the coarse and fine meshes in the first stage, which characterize the material distribution state of the discrete units within the unit cell. 1 represents solid material and 0 represents voids. Let be the objective function of the topology optimization model, representing the th Rank and number The relative band gap width between energy bands, where Indicates the first The maximum eigenfrequency of the first energy band, For the first The minimum eigenfrequency of the first energy band; Equivalent permeability is used to characterize the connectivity of single-phase materials. This is the penetration threshold.

[0141] It should be noted that:

[0142] The second stage involves gradient-based refinement optimization. While the first stage allowed design variables to take intermediate density values ​​to fully explore the design space, the resulting configurations did not meet engineering manufacturing requirements. Therefore, in this embodiment, the second stage selects high-performance candidate solutions from the Pareto front as initial input, and combines this with floating projected topology optimization (FPTO) and mesh refinement techniques (… The design unit is linearly encrypted. (Design unit) applies progressive 0 / 1 constraints to drive the design to evolve toward a manufacturable topology.

[0143] Where the objective function With the first objective function of the first stage Consistent, and equivalent penetration objective function The constraints are transformed to ensure the continuity and consistency of the two-stage topology optimization strategy. Post-processing methods are used to improve topology quality. First, a smooth design is extracted using the level set method, and then it is projected onto the original mesh while maintaining the equivalent volume, thereby refining the topology features and enhancing optimization convergence.

[0144] To further implement the above technical solution, Floating Projective Topology Optimization (FPTO), a robust gradient topology optimization method based on element density, uses gradient information to drive design variables to optimize specified physical properties. It transforms the discrete problem into a continuous problem through continuous relaxation, and simultaneously uses a linear interpolation model to define the relationship between element density and material properties: (essential step)

[0145] ;

[0146] In the formula: subscript This indicates parameters for solid materials and can be specified as needed. Unit density; and This represents the Lamé elastic constant.

[0147] Compared to the power-law penalized interpolation model, the linear interpolation model has advantages such as simpler formulas, higher computational efficiency, and no need to manually adjust the penalty factor. It is more applicable and provides simplified sensitivity analysis and improved numerical stability for large-scale computations.

[0148] To further implement the above technical solution, in the absence of the preset convergence condition, a three-field floating projection topology optimization framework is introduced to achieve topology optimization of the single-phase elastic superstructure. The specific content of updating the design field design variables includes:

[0149] If the preset convergence condition is not met, the design field design variables are converted into filter field design variables by density filtering, and then the filter field design variables are mapped to the physical field. The sensitivity of the design field design variables is obtained according to the chain rule, and the design field design variables are iteratively updated according to the sensitivity.

[0150] It should be noted that:

[0151] This invention introduces a three-field floating projection constraint topology optimization framework for bandgap optimization of single-phase elastic superstructures. This framework designs the field ( ), filtration field ( ) and physical fields ( The separation and coupling of the design variables in the design field achieve structural optimization. The design variables in the design field are iteratively updated, receiving gradient information from sensitivity analysis. The design variables in the filter field are transformed using density filtering technology to eliminate checkerboard patterns and ensure minimum feature size. The physical field, derived from the filter field variables through a projection function, is used for finite element calculation and evaluation of the objective function and constraints.

[0152] To further implement the above technical solution, the specific content of converting the design field design variables into filter field design variables through density filtering includes:

[0153] ;

[0154] in, Design variables for the filter field. Design variables and weighting factors for the design field , For the filter radius, For the first The and the first The center distance of each discrete unit Represents a vector matrix of discrete design variables The design variable index in the code.

[0155] It should be noted that:

[0156] In this embodiment, the filter radius is set to 2. This value can be set according to the number of grids and the specific problem.

[0157] To further implement the above technical solution, the specific content of mapping the filter field design variables to the physical field includes:

[0158] ;

[0159] In the formula, For the number of iterations, Indicates the first Physical field design variables in the next iteration process; For projection threshold, The projection sharpness parameter from the filter field to the physical field controls the degree of discretization of the variable. These are the floating projection parameters required for design variable iteration in the design field.

[0160] It should be noted that:

[0161] To eliminate intermediate density values ​​generated by density filtering, the Heaviside projection function is used to map the filter field to the physical field, achieving a clear 0-1 design characterization.

[0162] In this embodiment, the projection threshold Take 0.5, and >0.5 and <0.5 indicates the degree of erosion and expansion, respectively; By defining the projection sharpness parameter from the filter field to the physical field and controlling the degree of discretization of the control variables, this invention effectively controls topological complexity and manufacturability while ensuring numerical stability and optimization convergence through this three-field framework, providing a systematic technical solution for the efficient design of single-phase elastic superstructures.

[0163] To further implement the above technical solution, the specific content of the sensitivity of the design field design variables obtained according to the chain rule includes:

[0164] ;

[0165] In the formula, and The objective functions of the topology optimization model are respectively The sensitivity of the constraint function to the design variables of the design field;

[0166] The bandgap optimization problem is essentially a frequency optimization problem:

[0167] ;

[0168] In the formula, the first The partial derivative of the maximum value of the first-order eigenfrequency or the minimum value of the (n+1)th-order eigenfrequency with respect to the design variable is:

[0169] ;

[0170] In the formula, The normalized eigenmode vector; and These are the overall stiffness matrix and the mass matrix, respectively; The intrinsic frequency, For wave vectors;

[0171] To mitigate the numerical instability caused by discrete eigenfrequencys during optimization, an improved p-norm approximation method is used to transform the non-differentiable discrete eigenfrequencys on the irreducible Brillouin zone boundary into continuously differentiable functions:

[0172] ;

[0173] In the formula, the parameter It is a key factor controlling the approximate smoothness of the p-norm;

[0174] The sensitivity information for the equivalent permeability constraint is as follows:

[0175] ;

[0176] In the formula, Design domain volume; This represents the unit equivalent penetration rate.

[0177] It should be noted that:

[0178] The performance sensitivity information of the physical field is fed back to the design field step by step, providing precise directional guidance for the gradient update of the design variables.

[0179] In this embodiment, parameters The value is set to 8 to strike a balance between optimizing convergence and computational efficiency. This approximation method smooths the eigenfrequency, which was originally discretely sampled at the Brillouin zone boundary, thus avoiding the numerical singularity problem in gradient calculation.

[0180] The sensitivity information of the equivalent permeability constraint is used to determine and correct the constraint satisfaction in the gradient optimization algorithm, thereby providing a basis for the iterative update of the design variables.

[0181] To further implement the above technical solution, the specific content of iteratively updating the design variables of the design field based on sensitivity includes:

[0182] ;

[0183] in, This represents the current iteration number; and They are respectively The objective function of the topology optimization model at -1 iterations The sensitivity of the constraint function to the design variables of the design field; These are Lagrange multipliers used to reinforce the equivalent permeability constraint;

[0184] Furthermore, by introducing implicit floating projection constraints in the design field, the smooth and clear boundary representation is continuously refined and enhanced:

[0185] ;

[0186] Among them, projection threshold Conservation of the sum before and after projection of design variables is achieved through design variable projection. Implicit determination; The floating projection parameters required for design variable iteration in the design field;

[0187] Get Use an incremental approach: starting from the initial value Starting with =1, when the convergence criterion is satisfied... When the precision is less than or equal to the preset precision or the number of predefined iterations is reached, Increasing Δ =1.

[0188] It should be noted that:

[0189] During the optimization iteration process, when the physical topology has not yet met the specified penetration threshold... At that time, the iterative calculation of the Lagrange multipliers is in a paused state (i.e., (Keep it as the initial value or do not participate in the update), once the equivalent permeability of the structure... Reaching or exceeding the constraint threshold Lagrange multipliers then begin to participate in the iterative update, dynamically adjusting their values ​​through the bisection method or Newton's method to ensure that the constraints are continuously satisfied in subsequent iterations.

[0190] Since Floating Projective Topology Optimization (FPTO) uses a linear interpolation model instead of a traditional penalized power-law scheme, an additional mechanism is needed to drive the relaxed design variables to evolve into discrete 0 / 1 solutions to ensure equivalence with the original discrete optimization problem. This invention continuously refines and strengthens the smooth and clear boundary representation by introducing implicit floating projection constraints into the design field.

[0191] In this embodiment, obtain In the incremental approach, starting with a smaller initial value Starting with =1, when the convergence criterion is satisfied... Or when the predefined number of iterations reaches 60, Increasing Δ =1. This asymptotic strategy gradually transitions the optimization process from the initial continuous grayscale domain to the later discrete 0 / 1 domain, improving the quality of the solution while ensuring numerical stability.

[0192] In order to further implement the above technical solution, when After convergence, the final iterative convergence determination of the optimization algorithm is performed, i.e., the preset convergence condition is:

[0193] Unit solution By interpolating to the pixel scale using level set functions, the smoothed design is then projected back onto the original mesh while preserving the equivalent volume, resulting in a smoothed solution. ;

[0194] The relative performance difference between the smooth solution and the element solution is used to determine the convergence of the second-stage iteration:

[0195] ;

[0196] When the convergence condition is met, the smooth solution is considered equivalent to the element solution, and the corresponding optimized configuration is obtained, thus terminating the optimization.

[0197] Otherwise, continue iterating until the convergence condition is met or the maximum number of iterations is reached.

[0198] It should be noted that:

[0199] To accurately characterize the smooth boundaries of the optimized topology, a mesh refinement and reconstruction strategy is introduced. Finite elements are subdivided into dense pixels to improve boundary resolution, and the element solution... By interpolating to the pixel scale using level set functions, the smoothed design is then projected back onto the original mesh while preserving the equivalent volume, resulting in a smoothed solution. The relative performance difference between the two solutions (smooth solution and element solution) is used to determine the convergence of the second-stage iterative process.

[0200] When the convergence condition is met, the two solutions are considered equivalent, and their optimized configurations can be directly used for manufacturing, thus terminating the optimization; otherwise, iteration continues until the criterion is met or the maximum number of iterations is reached. Furthermore, this embodiment employs a moving limit factor. This allows for the adjustment of the magnitude of design variable updates in each iteration, and the application of upper and lower bound constraints improves numerical stability.

[0201] Through dynamic control of the floating projection constraints, mesh refinement and reconstruction, and performance equivalence verification, this invention achieves an efficient transformation from continuous relaxation solutions to discrete fabricable topologies. While ensuring optimization convergence and numerical stability, it obtains high-performance design configurations with smooth boundaries and clear 0 / 1 material distribution.

[0202] The invention will be further illustrated below through specific experiments:

[0203] This embodiment employs the real-number encoded NSGA-II algorithm for the first stage of global multi-objective search, optimizing the relative width of the in-plane bandgap and structural connectivity in the third order. The non-gradient topology optimization process diagram is shown below. Figure 2 As shown, the optimization results indicate that, with iteration, the proportion of Pareto front solutions in the population gradually increases from approximately 10% initially to 100%, eventually forming a continuous and uniformly distributed Pareto front, demonstrating the competitive relationship between the relative bandgap width and equivalent permeability. Representative solutions RUC-A1, RUC-A2, and RUC-A3 were selected from the Pareto front as the initial configurations for the second-stage fine optimization, providing high-quality candidate solutions for subsequent gradient optimization.

[0204] In the second stage, a three-field floating projection topology optimization method is used to refine the representative solution of the Pareto front. This is achieved through mesh refinement (from a 16×16 coarse mesh to a 64×64 fine mesh) and with the equivalent permeability of the selected solution as a constraint. Gradient iteration is then performed using the improved optimization criterion (OC) method. Based on the initial feasible solution with bandgap characteristics obtained in the first stage of topology optimization, further refinement is performed using the floating projection topology optimization method. The optimization process is as follows: Figure 3As shown, with the dynamic adjustment of the floating projection parameters, the material distribution gradually evolves from continuous grayscale to a clear 0 / 1 discrete topology. Optimization results show that this method can obtain a configuration with smooth boundaries and excellent bandgap characteristics in only 30-60 iterations, and the computational efficiency is improved compared with single non-gradient or gradient algorithms. All three optimized configurations exhibit consistent topological characteristics: concentrated mass blocks are connected by thin ligaments to form local resonant units, and quantitative analysis verifies the competitive relationship between bandgap characteristics and structural connectivity.

[0205] like Figure 4 As shown, the real band structure of the optimized configurations (RUC-A1, A2, A3) was verified using COMSOL software. The results show that all three configurations successfully opened and widened the bandgap between the 3rd and 4th order in-plane dispersion curves, but the relative width of the bandgap was inversely proportional to the structural connectivity. Intrinsic modal analysis revealed that the lower boundary of the bandgap exhibited symmetrical resonance of a concentrated mass block, while the upper boundary showed local vibration characteristics of a connecting ligament, verifying the important role of the concentrated mass block in the formation of the low-frequency bandgap.

[0206] In this embodiment, fused deposition modeling (FDM) was used to fabricate a 4×4 finite-period array experimental prototype (200×200×50mm, lattice constant 95A) composed of optimized unit cells RUC-A2 and RUC-A4 via 3D printing. =50mm). A layer of solid boundary elements is designed around the periodic array to facilitate excitation application and sensor placement. A homogeneous reference plate is fabricated simultaneously for comparison. A complete dynamic response testing system is constructed: a pulse generator (AWA1651) generates 0-4000Hz ( =0-1.2) frequency sweep excitation, amplified by a power amplifier (HEA-200C), is applied to the left side of the sample by a vibrator (HEV-200); high-precision accelerometers (LC-0101E) are placed at the input and output ends, and the signals are collected and the vibration transmissibility TR is calculated by a data acquisition module (B&K 3050-A). All samples are suspended to achieve free boundary conditions.

[0207] Depend on Figure 5 It can be seen that the accelerometer used in this invention exhibits excellent signal-to-noise ratio and sensitivity characteristics. Experimental results show that the homogeneous plate structure does not show a significant elastic wave attenuation effect in a specific frequency band; its vibration attenuation mainly stems from the inherent damping characteristics of the TPU flexible matrix material, and only exhibits a limited attenuation effect in the mid-to-high frequency range. In contrast, the finite-period arrays composed of the two-dimensional elastic superstructure configurations RUC-A2 and A4 designed using a hybrid optimization algorithm achieve attenuation in the dimensionless frequency range. =0.37~0.66, The bandgap characteristics exhibited were significant within the range of 0.46 to 0.94, which is highly consistent with the theoretically predicted bandgap characteristics of the band structure. The finite-period array with the optimized configuration exhibited significant vibration attenuation characteristics for the propagation of in-plane longitudinal and transverse waves within the bandgap frequency range. The experimental and simulation errors mainly stemmed from manufacturing deviations, differences in material properties, and the influence of boundary conditions; however, the two methods still showed a high degree of agreement in terms of attenuation frequency band, attenuation amplitude, and bandgap characteristics, fully verifying the effectiveness, reliability, and engineering applicability of the method presented in this invention.

[0208] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application, and should all be included within the protection scope of this application.

Claims

1. A non-gradient and gradient-coordinated topology optimization method for single-phase elastic superstructures, characterized in that, Includes the following steps: Phase 1: Complete NSGA-II multi-objective optimization; Generate a random initial configuration for the unit cell, and encode the random initial configuration and design variables using real numbers. The encoded design variable vector is denoted as... ; based on A multi-objective optimization model considering both maximizing bandgap performance and maximizing structural connectivity is constructed to transform the wave vector. The constraint is set on the scan path along the boundary of the irreducible Brillouin zone. This constraint is solved using a non-dominated sorting genetic algorithm to obtain... The Pareto optimal solution set converges to the Pareto front after population evolution; The multi-objective optimization model considering both maximizing bandgap performance and maximizing structural connectivity is as follows: ； ； In the formula, , The design variable vector represents the material distribution state of discrete elements within a unit cell, where 1 represents solid material and 0 represents voids. Indicates the first in the design domain Design variables for discrete units Indicates the total number of discrete units; Let be the first objective function, defined as the... Rank and number The relative band gap width between energy bands, where and They represent the first Rank and first The maximum eigenfrequency of the first energy band, and The first Rank and first The minimum eigenfrequency of the first energy band; The second objective function is defined as the equivalent permeability of the optimized structure. , used to characterize the connectivity of single-phase materials; The equivalent permeability of the discrete unit. It is a two-dimensional or three-dimensional identity matrix. For the design domain volume, It is a trial function, related to two-dimensional or three-dimensional problems; and They are respectively direction and The directional component is a homogenized variable. and They represent direction and The basis vector of direction, It is the unit cell lattice constant. It is a positive real number close to zero; Phase Two: Complete three FPTO topology optimizations; Select high-performance candidate solutions as initial inputs from the Pareto front; Based on high-performance candidate solutions, a topology optimization model is constructed that considers the maximization of bandgap performance and structural connectivity as constraints. The current design variables are solved. If the preset convergence condition is not met, a three-field floating projection topology optimization framework is introduced to realize the topology optimization of the single-phase elastic superstructure. The design field design variables are updated, and then implicit floating projection constraints are introduced to further iterate and update the design field design variables until the preset convergence condition is met.

2. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 1, characterized in that, The irreducible Brillouin zone (IBZ) corresponding to the unit cell is the minimal repeating region of the first Brillouin zone after symmetry operations; uniformly distributed wave vector sampling points are set along the boundary of the IBZ, and the scanning path is set to point M. → Point (0,0) → Point X → Point M The eigenfrequency corresponding to each wave vector sampling point is obtained, thereby obtaining the band dispersion curve. eigenfrequency Follow the wave arrow The changing relationship.

3. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 1, characterized in that, The problem was solved using a non-dominated sorting genetic algorithm, yielding... The specific contents of the Pareto optimal solution set include: (1) Initialize and set the maximum number of iterations Population size Number of discrete units Randomly generate the initial population Each individual All are 3D design variable vector; (2) For the current generation population Finite element analysis was performed on each individual in the population to calculate the objective function value; non-dominated ordination was performed based on Pareto dominance to divide the population into different levels of non-dominated fronts, and crowding distance of each individual was calculated. (3) A tournament selection strategy is used to select parent individuals from the current population, and the generation of the second generation is generated by simulating binary crossover (SBX) and polynomial mutation operations. Offspring population ; (4) Merging parent populations and offspring population Implement an elite retention strategy, selecting based on non-dominant level and crowding distance. The next generation of the population consists of outstanding individuals. ; (5) Repeat (2)-(4) until the maximum iteration number M is reached, and output the final Pareto front solution set, which contains a set of non-dominated optimal solutions that take into account both bandgap performance and structural connectivity.

4. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 1, characterized in that, The specific content of the topology optimization model considering bandgap performance maximization and structural connectivity as constraints is as follows: ； In the formula, The design variables for the second-stage gradient optimization algorithm are represented by the linear difference between the coarse and fine meshes in the first stage, which characterize the material distribution state of the discrete units within the unit cell. 1 represents solid material and 0 represents voids. Let be the objective function of the topology optimization model, representing the th Rank and number The relative band gap width between energy bands, where Indicates the first The maximum eigenfrequency of the first energy band, For the first The minimum eigenfrequency of the first energy band; Equivalent permeability is used to characterize the connectivity of single-phase materials. This is the penetration threshold.

5. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 4, characterized in that, In the FPTO process, dimensionless element design variables are interpolated using a material interpolation model. Mapped to physically meaningful material parameters: ； In the formula: subscript Indicates solid phase material parameters; For material density, and This represents the Lamé elastic constant.

6. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 4, characterized in that, In the absence of preset convergence conditions, a three-field floating projection topology optimization framework is introduced to achieve topology optimization of a single-phase elastic superstructure. The specific content of updating the design field design variables includes: If the preset convergence condition is not met, the design field design variables are converted into filter field design variables by density filtering, and then the filter field design variables are mapped to the physical field. The sensitivity of the design field design variables is obtained according to the chain rule, and the design field design variables are iteratively updated according to the sensitivity.

7. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 6, characterized in that, The specific content of converting design field design variables into filter field design variables through density filtering includes: ； in, Design variables for the filter field. Design variables and weighting factors for the design field , For the filter radius, For the first The and the first The center distance of each discrete unit Represents a vector matrix of discrete design variables The design variable index in the code.

8. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 7, characterized in that, The specific content of mapping filter field design variables to physical fields includes: ； In the formula, For the number of iterations, Indicates the first Physical field design variables in the next iteration process; For projection threshold, The projection sharpness parameter from the filter field to the physical field controls the degree of discretization of the variable. These are the floating projection parameters required for design variable iteration in the design field.

9. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 8, characterized in that, The specific details of obtaining the sensitivity of the design variables for the design field according to the chain rule include: ； In the formula, and The objective functions of the topology optimization model are respectively The sensitivity of the constraint function to the design variables of the design field; No. The partial derivatives of the first-order intrinsic frequencies with respect to the design variables are: ； In the formula, the first The partial derivative of the maximum value of the first-order eigenfrequency or the minimum value of the (n+1)th-order eigenfrequency with respect to the design variable is: ； In the formula, The normalized eigenmode vector; and These are the overall stiffness matrix and the mass matrix, respectively; The intrinsic frequency, For wave vectors; ； In the formula, parameter s is the key factor controlling the approximate smoothness of the p-norm; ； In the formula, Design domain volume; This represents the unit equivalent penetration rate.

10. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 4, characterized in that, The specific content of iteratively updating the design variables of the design field based on sensitivity includes: ； in, This represents the current iteration number; and They are respectively The objective function of the topology optimization model at -1 iterations The sensitivity of the constraint function to the design variables of the design field; These are Lagrange multipliers used to reinforce the equivalent permeability constraint; Furthermore, by introducing implicit floating projection constraints in the design field, the smooth and clear boundary representation is continuously refined and enhanced: ； Among them, projection threshold Conservation of the sum before and after projection of design variables is achieved through design variable projection. Implicit determination; The floating projection parameters required for design variable iteration in the design field; Get Use an incremental approach: starting from the initial value Starting with =1, when the convergence criterion is satisfied... When the precision is less than or equal to the preset precision or the number of predefined iterations is reached, Increasing .

11. The non-gradient and gradient cooperative topology optimization method for a single-phase elastic superstructure according to claim 10, characterized in that, Once β converges, the final iterative convergence determination of the optimization algorithm is performed, i.e., the preset convergence condition is: Unit solution By interpolating to the pixel scale using level set functions, the smoothed design is then projected back onto the original mesh while preserving the equivalent volume, resulting in a smoothed solution. ; The relative performance difference between the smooth solution and the element solution is used to determine the convergence of the second-stage iteration: ； When the convergence condition is met, the smooth solution is considered equivalent to the element solution, and the corresponding optimized configuration is obtained, thus terminating the optimization. Otherwise, continue iterating until the convergence condition is met or the maximum number of iterations is reached.

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