Two-dimensional photonic crystal multi-target topological optimization method based on gradient method

Abstract

The invention discloses a gradient method-based two-dimensional photonic crystal multi-target topological optimization method, which comprises the following steps of: constructing a finite element analysis model of a photonic crystal, taking the relative density of a unit as a design variable, and parameterizing material attributes by adopting a material interpolation model; discrete characteristic frequency data are acquired and smoothed; constructing a multi-objective optimization problem with the objective of simultaneously maximizing the band gap width and minimizing the structural flexibility, or simultaneously maximizing the band gap width and minimizing the thermal flexibility; respectively calculating the sensitivity of each target function to the design variable, and carrying out normalization processing; carrying out weighted fusion on the normalized sensitivity; filtering and historically averaging the sensitivity field to smooth the update process; a clear topological structure is obtained by using Heaviside projection; updating design variables based on an optimal criterion method; and iteratively optimizing until a preset convergence criterion is met, and outputting an optimal topological configuration. The design efficiency of the comprehensive performance of the photonic crystal is remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of phononic crystal topology optimization technology, specifically to a multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method. Background Technology

[0002] Phononic crystals are artificial materials with periodic structures that can modulate the propagation behavior of elastic waves, finding wide applications in vibration suppression, acoustic wave conduction, and acoustic stealth. Traditional phononic crystal design often relies on experience or trial-and-error methods, making it difficult to simultaneously ensure bandgap performance while maintaining other physical properties such as structural stiffness or thermal conductivity. With the increasing demands for multifunctional materials in engineering applications, achieving synergistic optimization between wide bandwidth and good mechanical / thermal properties has become a critical issue that urgently needs to be addressed.

[0003] Topology optimization can achieve optimal configuration of structural performance by optimizing the spatial distribution of materials within the design domain. However, the bandgap characteristics of phononic crystals are intricately coupled with objectives such as structural compliance and thermal compliance, making it difficult for traditional single-objective optimization methods to achieve a balanced design of multiple physical properties. Furthermore, the non-smoothness of the bandgap objective function and the numerical instability in the multi-objective sensitivity integration process also limit the further application of topology optimization in the collaborative design of multiple properties of phononic crystals.

[0004] While some existing research attempts to apply topology optimization to phononic crystal design, most focuses on optimizing single bandgap performance, lacking a comprehensive consideration of multi-physics performance. For example, phononic crystal structures in aerospace applications require excellent vibration isolation performance (wide bandgap), sufficient structural stiffness (low structural flexibility) to withstand mechanical loads, or good heat dissipation performance (low thermal flexibility) to prevent overheating. Simply pursuing a wider bandgap often leads to insufficient structural stiffness or deteriorated thermal performance, and vice versa. The lack of a topology optimization framework that can effectively and synergistically optimize the bandgap and multiple mechanical / thermal properties of phononic crystals results in poor overall performance of the designed structures, making them difficult to apply directly to complex multi-physics environments. Furthermore, traditional multi-objective optimization methods often employ fixed-weight strategies, making it difficult to flexibly adapt to the diverse needs of different engineering scenarios. Therefore, there is an urgent need to develop a method for efficient and stable multi-objective topology optimization of phononic crystals to meet the design needs of modern engineering for multifunctional materials. Summary of the Invention

[0005] To overcome the defects and shortcomings of existing technologies, this invention provides a multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method. This invention constructs a multi-objective topology optimization framework for phononic crystals with multiple physical properties, and simultaneously optimizes the elastic wave bandgap and structural / thermal compliance of phononic crystals through gradient optimization algorithms, which significantly improves the design efficiency of the overall performance of phononic crystals.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: This invention provides a multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method, comprising the following steps: A finite element analysis model of a phononic crystal is constructed, with the relative density of the elements as the design variable, and the material properties are parameterized using a material interpolation model. Obtain discrete characteristic frequency data and smooth the discrete characteristic frequencies; Construct a multi-objective optimization problem, the objective of which is to simultaneously maximize the bandgap width and minimize the structural compliance, or simultaneously maximize the bandgap width and minimize the thermal compliance. Calculate the sensitivity of each objective function to the design variables, and normalize the sensitivity sequence of each objective function. The normalized sensitivity is weighted and fused using preset weighting coefficients to form a unified comprehensive sensitivity field; The sensitivity field is filtered and historically averaged to smooth the update process, and Heaviside projection is used to obtain a clear topology. Update design variables based on the optimal criterion method; The system iterates through a predetermined convergence criterion until the criterion is met, outputting the optimal topology configuration.

[0007] As a preferred technical solution, the finite element analysis model includes the wave equation finite element model, as well as the finite element model for structural mechanics analysis or steady-state heat conduction analysis.

[0008] As a preferred technical solution, a finite element analysis model of a phononic crystal is constructed, specifically including: A two-dimensional phononic crystal unit cell design domain is defined, a two-phase material system is adopted, and a material volume fraction constraint is set. Based on Bloch's theorem, a wave equation finite element model under periodic boundary conditions is established. The adjacent band gap numbers to be optimized are selected, and the boundary conditions and loads required for structural compliance or thermal compliance optimization are set. At the same time, a finite element model for structural mechanical analysis or steady-state heat conduction analysis is established.

[0009] As a preferred technical solution, a material interpolation model is used to parameterize material properties, specifically including: The material interpolation model uses linear interpolation in bandgap calculation; An exponential penalty model based on the elastic modulus is used in structural flexibility analysis. An exponential penalty model for thermal conductivity is used in thermal flexibility analysis.

[0010] As a preferred technical solution, the material properties include Lamé coefficient, elastic modulus, mass density, and thermal conductivity.

[0011] As a preferred technical solution, smoothing the discrete characteristic frequencies specifically includes: The KS function is used to smooth discrete characteristic frequencies. When processing the lower bandgap frequency, the KS function adopts a lower bound aggregation form, and when processing the upper bandgap frequency, it adopts an upper bound aggregation form.

[0012] As a preferred technical solution, the sensitivity of each objective function to the design variables is calculated separately, specifically including: The sensitivities of the bandgap target, structural flexibility target, and thermal flexibility target to the design variables were calculated separately. The bandgap sensitivity was solved using an analytical method based on eigenvalue derivatives, while the structural flexibility sensitivity and thermal flexibility sensitivity were solved using the adjoint variable method.

[0013] As a preferred technical solution, the sensitivity sequences of each objective function are normalized, specifically including: The original bandgap sensitivity value is subjected to a base-10 logarithmic transformation, and the transformed value is mapped to the interval using the maximum-minimum normalization method. The structural flexibility sensitivity and thermal flexibility sensitivity are mapped using the maximum-minimum normalization method.

[0014] As a preferred technical solution, the optimal criterion method introduces a movement constraint mechanism during the design variable update process. By setting a damping coefficient, the maximum change of the design variable in a single iteration is limited. The damping coefficient is determined according to the nonlinearity of the optimization problem.

[0015] As a preferred technical solution, the sensitivity field is filtered using a linear filtering method based on convolution operations, with the filtering radius and cell size maintaining a fixed ratio, and the weighting function determined based on the distance between cell centers. Heaviside projection uses a continuously differentiable approximate step function, and controls the sharpness of the material boundaries by adjusting the projection parameters.

[0016] Compared with the prior art, the present invention has the following advantages and beneficial effects: (1) By constructing a multi-objective topology optimization framework for phononic crystals, this invention realizes the coordinated automatic design of phononic crystal bandgap characteristics and structural / thermal performance. Compared with the traditional single-objective optimization method, this invention significantly improves the design efficiency of the comprehensive performance of phononic crystals.

[0017] (2) Based on the intrinsic relationship between multiple physical properties of phononic crystals, this invention establishes a unified optimization framework. By adjusting the weighting factor, it achieves a flexible trade-off between different performance objectives, making the optimization process closely related to the physical nature of each performance objective, thereby broadening the scope of application of the method.

[0018] (3) This invention uses the KS function to process the non-smooth characteristics of the bandgap objective function and combines the sensitivity normalization method to handle the order-of-magnitude differences in the sensitivity of different physical fields. This effectively overcomes the numerical difficulties in multi-objective topology optimization and improves the stability and convergence efficiency of the optimization process.

[0019] (4) The present invention adopts a technical solution that integrates sensitivity filtering, historical averaging strategy and Heaviside projection technology in optimization iteration, which solves the technical problems of checkerboard phenomenon, numerical oscillation and material boundary ambiguity in topology optimization, and ensures the clarity, manufacturability and good physical realizability of the final topology configuration.

[0020] (5) This invention demonstrates the feasibility and effectiveness of synergistic optimization of bandgap-structural compliance and bandgap-thermal compliance under different weight combinations through multiple embodiments, solves the technical problem of lack of reliable and systematic method support in the multifunctional integrated design of phononic crystals, and provides a systematic, flexible and stable optimization tool for the design of multifunctional phononic crystals. Attached Figure Description

[0021] Figure 1 This is a flowchart illustrating the multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method of this invention. Figure 2(a) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 2(b) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 2(c) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 2(d) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 2(e) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 3 This is the optimization iteration history diagram of Figure 2(d); Figure 4(a) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 4(b) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 4(c) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 4(d) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 4(e) shows the weighting coefficients. , Optimized structure diagram and bandgap diagram at the time; Figure 5 This is the optimization iteration history diagram of Figure 4(d). Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0023] Example 1 like Figure 1 As shown, this embodiment provides a multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method, including the following steps: S1: Construct a phononic crystal finite element analysis model, using the relative density of the elements as the design variable, and use a material interpolation model to parameterize the material properties, including Lamé coefficient, elastic modulus, mass density and thermal conductivity. In this embodiment, the specific problems of constructing the finite element analysis model of the phononic crystal include: setting a two-dimensional phononic crystal unit cell design domain, adopting a two-phase material system, setting material volume fraction constraints, establishing a wave equation finite element model under periodic boundary conditions based on Bloch's theorem, selecting the adjacent bandgap numbers to be optimized; setting the boundary conditions and loads required for structural compliance or thermal compliance optimization, and simultaneously establishing a finite element model for structural mechanical analysis or steady-state heat conduction analysis. In this embodiment, the two-dimensional phononic crystal unit cell design domain adopts a 1×1 unit length square lattice, discretized into 32×32 four-node quadrilateral elements. The two-phase material system consists of two isotropic materials: material 1 has a density of 1000 kg / m³, a first Lamé parameter of 5.77 GPa, and a second Lamé parameter of 3.85 GPa; material 2 has a density of 8000 kg / m³, a first Lamé parameter of 577 GPa, and a second Lamé parameter of 385 GPa. The volume fraction constraint of material 2 is set to 0.4. The Heaviside projection function is used to promote a clear 0-1 material distribution. The projection parameter starts from an initial value of 1 and doubles every 20 iterations, increasing to a maximum of 128. The boundary conditions for structural flexibility optimization are specifically set as follows: a vertically downward concentrated unit load is applied to the upper left node of the unit cell; all nodes on the left boundary of the unit cell are constrained to displacement in the X direction; the lower right node of the unit cell is completely fixed, constraining all degrees of freedom.

[0024] In this embodiment, a finite element model of the elastic wave control equation under periodic boundary conditions is established based on Bloch's theorem, and the displacement vector can be expressed as: ,in yes A periodic function whose periodicity and structure are the same. Let represent the Bloch wave vector. 31 wave vector points were precisely sampled at the boundary of the irreducible Brillouin zone, among which... 10 points in the direction, 10 points in the direction, There are 11 points in the direction. The bandgap calculation uses the KS function to aggregate the characteristic frequencies; the objective function can be expressed as... , The characteristic frequency of a single cell is denoted by , and n represents the number of points obtained from the irreducible Brillouin zone. The parameters of the KS function are set to . , Simultaneously, establish a finite element model for structural mechanics or steady-state heat conduction analysis: (1) Calculate the structural flexibility, the objective function of which is defined as ,in The global stiffness matrix. To solve the equilibrium equations The obtained displacement vector, For the load vector. (2) Calculate the thermal compliance, the objective function of thermal compliance is defined as ,in The global stiffness matrix. To solve the equilibrium equations The obtained displacement vector, This is the heat-carrying vector.

[0025] In this embodiment, the material interpolation model uses linear interpolation in the bandgap calculation. The interpolation method can be expressed as follows: ,in Indicates density, This represents the first Lamé coefficient. This represents the second Lamé coefficient, with the subscript indicating material 1 or material 2. In structural flexibility analysis, an exponential penalty model based on the elastic modulus is used, and the interpolation method can be expressed as... In thermal flexibility analysis, an exponential penalty model for thermal conductivity is used, and the interpolation method can be expressed as follows: The penalty factor p is adjusted according to the requirements of numerical stability. S2: Sample multiple wave vector points along a set path at the boundary of the irreducible Brillouin zone, solve the eigenvalue problem corresponding to the elastic wave control equation, and obtain discrete characteristic frequency data; S3: The KS (Kreisselmeier-Steinhauser function) is used to smooth the discrete feature frequencies to effectively solve the non-differentiability problem caused by the switching of the characteristic frequency order during the optimization process. Based on the KS function, the discrete feature frequencies are aggregated to construct a continuously differentiable bandgap width objective function. In this embodiment, the KS function uses a lower bound aggregation method when processing the lower bandgap frequency and an upper bound aggregation method when processing the upper bandgap frequency. The approximation accuracy is controlled by adjusting the aggregation parameters, wherein the selection of the aggregation parameters takes into account the numerical range and distribution characteristics of the characteristic frequencies. S4: Construct a multi-objective function whose objective is to simultaneously maximize the bandgap width and minimize the structural compliance, or simultaneously maximize the bandgap width and minimize the thermal compliance. In this embodiment, a multi-objective function based on the weighted sum method is constructed, expressed as: ; in, Given the bandgap objective function, the KS function is used to aggregate the characteristic frequencies. For structural flexibility or thermal flexibility , and It is a weighting factor, and satisfies ; In this embodiment, to comprehensively explore the design space, five representative combinations of weighting factors are set: ( , ), ( , ), ( , ), ( , ), ( , ).

[0026] S5: Calculate the sensitivity of each objective function to the design variables, and perform two-stage normalization on the sensitivity sequence of each objective function. In this embodiment, the sensitivities of the bandgap target and the structural and thermal flexibility targets to the design variables are calculated separately. The bandgap sensitivity is solved using an analytical method based on eigenvalue derivatives, while the structural and thermal flexibility sensitivities are solved using the adjoint variable method. Furthermore, a logarithmic transformation combined with a minimum-maximum normalization method is used to process sensitivity values ​​of different orders of magnitude. The bandgap sensitivity is solved precisely using an analytical method based on eigenvalue derivatives, specifically in the form of: ; The structural flexibility sensitivity is solved using the adjoint variable method, in the form of: , Represents the element displacement vector. Let represent the element stiffness matrix of material 2. Material interpolation uses the SIMP model, and the elastic modulus interpolation formula is: Punishment factor .

[0027] The thermal flexibility sensitivity is also solved using the adjoint variable method, in the form of: , Represents the element temperature vector. This represents the unit thermal conductivity matrix of material 2. Material interpolation uses the SIMP model, and the elastic modulus interpolation formula is: Punishment factor .

[0028] In this embodiment, the two-stage normalization process of the sensitivity sequence includes two stages: logarithmic transformation and minimum-maximum normalization. The logarithmic transformation is used to compress the dynamic range of the sensitivity, and the minimum-maximum normalization is used to map the transformed sensitivity to a standard numerical range. In this embodiment, the original bandgap sensitivity value is subjected to a base-10 logarithmic transformation. The transformed value is then mapped to an interval using a maximum-minimum normalization method, effectively eliminating the order-of-magnitude differences in sensitivity across different physical fields. Structural flexibility is directly mapped using the maximum-minimum normalization method. An interval-based sensitivity filtering scheme is used, with the filtering radius set to [value missing]. A bilinear weighting function is used to ensure filtering effectiveness. Starting from the second iteration, a historical averaging strategy is applied, with the mixing coefficient between the current sensitivity and historical sensitivity set to 0.5 to smooth the update process. The optimal criterion method is used to update the design variables, with the moving average limit set to 0.05. The Lagrange multipliers are used to accurately solve for volume constraints using the bisection method.

[0029] S6: The normalized sensitivity is weighted and fused using preset weighting coefficients to form a unified comprehensive sensitivity field; S7: Update design variables and perform iterative optimization, specifically including: fusing the normalized sensitivities according to weighting factors, updating design variables based on the comprehensive sensitivity field using the optimal criterion method, and the specific update method is as follows. ,in , For the weighted objective function, For Lagrange multipliers, The value is a volume function. Sensitivity filtering, historical averaging, and Heaviside projection techniques are combined to ensure the numerical stability of the optimization process. In this embodiment, the optimal criterion method introduces a movement constraint mechanism during the design variable update process by setting a damping coefficient. To limit the maximum variation of design variables in a single iteration, the damping coefficient is determined based on the nonlinearity of the optimization problem; In this embodiment, the sensitivity filtering adopts a linear filtering method based on convolution operation, the filtering radius is kept in a fixed ratio with the cell size, and the weight function is determined according to the distance between the cell centers; In this embodiment, the projection method uses a continuously differentiable approximate step function. The clarity of the material boundary is controlled by adjusting the projection parameters, and the projection parameters can be adaptively adjusted during the optimization process. S8: Monitor the relative change history of the combined objective function and determine whether to terminate the optimization process based on the convergence criterion.

[0030] In this embodiment, the convergence criterion adopts the judgment standard based on the relative change of the objective function, and comprehensively considers the changes of the design variables. When the preset convergence criterion is met, the optimization process is terminated and the optimal topology with a clear material interface is output. This embodiment sets a dual convergence criterion: optimization stops when the relative change of the objective function is less than a set threshold or when the maximum number of iterations is reached. This embodiment is applicable to optimizing the band gap generated between multiple adjacent energy bands by in-plane waves in phononic crystals, wherein the selection of the target energy band is determined according to the actual engineering application requirements.

[0031] In this embodiment, the two constituent materials of the phononic crystal have significant differences in elastic modulus, mass density and thermal conductivity parameters. The contrast between the material properties affects the band gap formation range and optimization results, and the phononic crystal unit cell is discretized using a regular quadrilateral grid. This invention effectively solves the common performance trade-off problem in multiphysics design through a gradient-driven topology optimization method. It has important application value in fields such as aerospace and automotive industries that have high requirements for integrated structural-thermal-acoustic functional materials. Furthermore, by integrating a multi-objective optimization process, it significantly improves the overall performance design efficiency of phononic crystals under multi-frequency conditions.

[0032] Example 2 This embodiment provides a multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method, demonstrating the multi-objective topology optimization design of the seventh bandgap (between the seventh and eighth bands) and structural compliance, specifically including the following steps: S1: Establish optimization problems and initial settings; A two-dimensional phononic crystal unit cell design domain is defined, using a 1×1 unit length cubic lattice, discretized into 32×32 four-node quadrilateral elements. The material system consists of two isotropic materials: Material 1 has a density of 1000 kg / m³, a first Lamé parameter of 5.77 GPa, and a second Lamé parameter of 3.85 GPa; Material 2 has a density of 8000 kg / m³, a first Lamé parameter of 577 GPa, and a second Lamé parameter of 385 GPa. The volume fraction constraint of Material 2 is set to 0.4. The Heaviside projection function is used to promote a clear 0-1 material distribution. The projection parameter starts from an initial value of 1 and doubles every 20 iterations, increasing to a maximum of 128. The boundary conditions for structural flexibility optimization are specifically set as follows: a vertically downward concentrated unit load is applied to the upper left node of the unit cell; all nodes on the left boundary of the unit cell are constrained in the X-direction displacement; the lower right node of the unit cell is fully fixed, constraining all degrees of freedom.

[0033] S2: Constructing the finite element analysis model and performance calculation; A finite element model of the wave equation under periodic boundary conditions was established based on Bloch's theorem. Thirty-one wave vector points were precisely sampled at the irreducible Brillouin zone boundary. 10 points in the direction, 10 points in the direction, The bandgap is calculated using the KS function to aggregate the characteristic frequencies across 11 points. The objective function can be expressed as follows: The parameters of the KS function are set to , Simultaneously, a structural mechanics analysis model is established to calculate the structural flexibility. The objective function for structural flexibility is defined as follows: ,in The global stiffness matrix. To solve the equilibrium equations The obtained displacement vector, For load vectors; S3: Set up multiple objective functions and sensitivity analysis, adopt a weighted average strategy, and integrate the sensitivity information of multiple objective functions such as phononic crystal bandgap width, structural compliance and thermal compliance; Constructing the comprehensive objective function using the weighted sum method ,in For the seventh bandgap objective function, To ensure structural flexibility and to fully explore the design space, five representative combinations of weighting factors were established: ( , ), ( , ), ( , ), ( , ), ( , The bandgap sensitivity is precisely solved using an analytical method based on eigenvalue derivatives, specifically in the form of... The structural flexibility sensitivity is solved using the adjoint variable method, in the form of: Material interpolation uses the SIMP model, and the elastic modulus interpolation formula is: Punishment factor .

[0034] S4: Sensitivity processing and design variable update optimization; The original bandgap sensitivity values ​​are subjected to a base-10 logarithmic transformation. The transformed values ​​are then mapped to intervals using a maximum-minimum normalization method, effectively eliminating the order-of-magnitude differences in sensitivity across different physical fields. Structural flexibility sensitivity and thermal flexibility sensitivity are directly mapped using the maximum-minimum normalization method. An interval-based sensitivity filtering scheme is employed, with the filtering radius set to [value missing]. A bilinear weighting function is used to ensure the filtering effect. The historical averaging strategy is applied starting from the second iteration. The mixing coefficient of the current sensitivity and the historical sensitivity is set to 0.5 to smooth the update process. Heaviside projection is used to obtain clear topological results. The optimal criterion method is used to update the design variables. The moving limit is set to 0.05. The Lagrange multiplier is used to solve the volume constraint accurately by the bisection method. S5: Convergence Judgment and Optimization Result Analysis: The design variables are updated based on the optimal criterion method, and the convergence condition is judged based on the relative change of the combined objective function. Through iterative optimization, a phononic crystal configuration with both wide bandwidth characteristics and excellent multi-physics performance is finally obtained. As shown in Figures 2(a)-2(e), this embodiment obtains a comparison of the final topological configuration and performance of the seventh bandgap under five different weighting factors. Specifically, the weights are... and At that time, the structure achieved a relative bandgap width of 34.86%, while the structural flexibility was improved compared to single-objective optimization. , The improvement was approximately 40%, and about 50% better than the initial design. The optimization results under other weight combinations also met the expected goals, successfully achieving an effective balance between wide bandwidth and high stiffness.

[0035] This embodiment sets a dual convergence criterion: when the relative change in the objective function is less than a threshold... Or stop optimization when the maximum number of iterations (200) is reached, when the weight coefficients... , The optimization iteration history at that time is as follows Figure 3 As shown, optimization typically reaches a stable state after 70-150 iterations.

[0036] Example 3 This embodiment demonstrates the multi-objective topology optimization design process for the fifth band gap (between the fifth and sixth bands) and thermal compliance. Except for the following technical contents, the optimization process and parameter settings in this embodiment are the same as those in Embodiment 2: In this embodiment, the performance objective in the problem-solving step is changed to the fifth bandgap and thermal compliance. The thermal boundary conditions are set to apply zero-temperature boundary conditions at the four corners of the unit cell and a unit heat source at the center of the unit cell. The material thermal conductivity is set to that of material 1. Material 2 is Thermal flexibility is defined as the thermal conductivity performance index of a system under given thermal boundary conditions. The smaller the value, the better the heat dissipation performance.

[0037] In this embodiment, the finite element analysis model construction step replaces structural mechanics analysis with steady-state heat conduction analysis, and the objective function is: The governing equation is ,in For the global heat conduction matrix, For the temperature field, The heat source is used. The material interpolation model for the heat conduction problem adopts an exponential penalty model based on thermal conductivity, specifically in the form of... Punishment factor Set it to 3.

[0038] In this embodiment, the multi-objective function construction step tests five different combinations of weight factors: ( , ), ( , ), ( , ), ( , ), ( , To comprehensively study the trade-off between the characteristics of the fifth bandgap and thermal performance.

[0039] In this embodiment, the sensitivity analysis step uses the adjoint method based on the temperature field to solve for the thermal compliance target. The specific derivation process differs significantly from the structural compliance sensitivity. The sensitivity expression for the thermal compliance target function is as follows: ,in For unit temperature matrix, This is the unit heat conduction matrix.

[0040] As shown in Figures 4(a)-4(e), the optimization results show significant differences in the final topology configurations under the five weight groups. When the weight is... , At the same time, the optimized structure achieved a relative bandgap width of 35.00% while maintaining low thermal flexibility, compared to , This set of weights resulted in a five-order-of-magnitude reduction, which is also five orders of magnitude lower than the initial design, demonstrating excellent overall acoustic-thermal performance. Of particular note is that when the weights are... , When performing single-objective optimization, although the largest relative bandgap of 48.66% was obtained, its thermal flexibility deteriorated significantly, resulting in the worst heat dissipation performance. When the weighting coefficient is... and The optimization iteration history at that time is as follows Figure 5 As shown, the optimization process has good convergence and stability. These results fully demonstrate that by adjusting the weighting factor, an effective trade-off can be made between the fifth bandgap characteristics and heat dissipation performance, providing a flexible design method for phononic crystal design in different application scenarios.

[0041] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A multi-objective topology optimization method for two-dimensional phonon crystals based on the gradient method, characterized in that, Includes the following steps: A finite element analysis model of a phononic crystal is constructed, with the relative density of the elements as the design variable, and the material properties are parameterized using a material interpolation model. Obtain discrete characteristic frequency data and smooth the discrete characteristic frequencies; Construct a multi-objective optimization problem, the objective of which is to simultaneously maximize the bandgap width and minimize the structural compliance, or simultaneously maximize the bandgap width and minimize the thermal compliance. Calculate the sensitivity of each objective function to the design variables, and normalize the sensitivity sequence of each objective function. The normalized sensitivity is weighted and fused using preset weighting coefficients to form a unified comprehensive sensitivity field; The sensitivity field is filtered and historically averaged to smooth the update process, and Heaviside projection is used to obtain a clear topology. Update design variables based on the optimal criterion method; The system iterates through a predetermined convergence criterion until the criterion is met, outputting the optimal topology configuration.

2. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 1, characterized in that, Finite element analysis models include wave equation finite element models, as well as finite element models for structural mechanics analysis or steady-state heat conduction analysis.

3. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 2, characterized in that, Constructing a finite element analysis model for phononic crystals, specifically including: A two-dimensional phononic crystal unit cell design domain is defined, a two-phase material system is adopted, and a material volume fraction constraint is set. Based on Bloch's theorem, a wave equation finite element model under periodic boundary conditions is established. The adjacent band gap numbers to be optimized are selected, and the boundary conditions and loads required for structural compliance or thermal compliance optimization are set. At the same time, a finite element model for structural mechanical analysis or steady-state heat conduction analysis is established.

4. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 1, characterized in that, Material properties are parameterized using a material interpolation model, specifically including: The material interpolation model uses linear interpolation in bandgap calculation; An exponential penalty model based on the elastic modulus is used in structural flexibility analysis. An exponential penalty model for thermal conductivity is used in thermal flexibility analysis.

5. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 4, characterized in that, Material properties include Lamé coefficient, elastic modulus, mass density, and thermal conductivity.

6. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 1, characterized in that, Smoothing of discrete characteristic frequencies specifically includes: The KS function is used to smooth discrete characteristic frequencies. When processing the lower bandgap frequency, the KS function adopts a lower bound aggregation form, and when processing the upper bandgap frequency, it adopts an upper bound aggregation form.

7. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 1, characterized in that, Calculate the sensitivity of each objective function to the design variables, specifically including: The sensitivities of the bandgap target, structural flexibility target, and thermal flexibility target to the design variables were calculated separately. The bandgap sensitivity was solved using an analytical method based on eigenvalue derivatives, while the structural flexibility sensitivity and thermal flexibility sensitivity were solved using the adjoint variable method.

8. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 7, characterized in that, The sensitivity sequences of each objective function are normalized, specifically including: The original bandgap sensitivity value is subjected to a base-10 logarithmic transformation, and the transformed value is mapped to the [0, 1] interval using the maximum-minimum normalization method; The structural flexibility sensitivity and thermal flexibility sensitivity are mapped to [0, 1] using the maximum-minimum normalization method.

9. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 1, characterized in that, The optimal criterion method introduces a movement constraint mechanism during the design variable update process. By setting a damping coefficient, the maximum change of the design variable in a single iteration is limited. The damping coefficient is determined according to the nonlinearity of the optimization problem.

10. The multi-objective topology optimization method for two-dimensional phononic crystals based on the gradient method according to claim 1, characterized in that, The sensitivity field is filtered using a linear filtering method based on convolution operations. The filtering radius is kept in a fixed ratio with the cell size, and the weighting function is determined based on the distance between the cell centers. Heaviside projection uses a continuously differentiable approximate step function, and controls the sharpness of the material boundaries by adjusting the projection parameters.

Images