A medium-high frequency thermal vibration structure topology optimization method based on variable density method

Abstract

The application discloses a kind of high-frequency thermal vibration structure topology optimization method based on variable density method, first establish thermal vibration coupling analysis model, energy finite element method is used to solve high-frequency vibration energy response, and the equivalent performance parameters of structure under the action of thermal vibration coupling are obtained by combining temperature field analysis;Based on variable density method, construct topology optimization model, minimize high-frequency vibration energy flexibility as optimization goal, integrate material density interpolation and penalty mechanism, deduce coupled field optimization sensitivity;Optimal distribution of material in design domain is realized by iteration optimization, and finally the optimal topology configuration of high-frequency structure meeting the requirements of lightweight, vibration reduction of thermal vibration coupling is obtained;The application realizes the deep integration of energy finite element high-frequency analysis and variable density method topology optimization in thermal vibration environment, improves the efficiency of thermal vibration coupling high-frequency structure optimization, adapts to the severe design requirements of key components in aerospace, high-end equipment and other fields, and has good engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of high-frequency dynamic performance optimization design of structures under thermal conditions, specifically involving a topology optimization method for medium- and high-frequency thermal vibration structures based on the variable density method. Background Technology

[0002] With the rapid development of industries such as aerospace, high-end equipment, and precision transmission, the service conditions of structures are becoming increasingly harsh. The coupling effect of high-temperature environment and high-frequency vibration has become a core factor restricting the reliability and lightweighting of equipment. Key components such as aero-engine combustion chambers, high-speed aircraft control surfaces, and precision servo mechanisms are simultaneously subjected to severe temperature field loads and broadband high-frequency vibration excitation during operation. Temperature-induced degradation of material elastic modulus and accumulation of thermal stress are coupled with high-frequency vibration energy transfer and structural resonance failure, which can easily lead to structural fatigue failure, excessive vibration response, and thermal-vibration coupling instability. Topology optimization design for such thermal-vibration coupled high-frequency structures to achieve optimal material distribution and synergistic improvement of multi-physics performance has become a key technical requirement in the field of modern structural design.

[0003] In the field of high-frequency vibration analysis of structures, the classical finite element method (FEA) must adhere to the wavelength discretization criterion, as exemplified by the patent application titled "Dynamic Calculation Method Based on Spectral Feature Encoding Network" (Publication No.: CN120974954A). While this method can reduce the number of iterations and improve solution efficiency, providing an efficient solution approach for high-frequency dynamic problems in engineering, it still requires a massive number of mesh elements to ensure solution accuracy in the high-frequency band, resulting in an exponential increase in computational cost and making it unsuitable for iterative solutions of large-scale topology optimization. Statistical energy analysis (SEA), although applicable to high-frequency systems (e.g., the patent application titled "A Method for Calculating the High-Frequency Dynamic Response of an Acoustic-Vibration System" (Publication No.: CN102411673A), can only obtain the average response of the subsystem and cannot characterize the local energy distribution and topological details of the structure, making it difficult to match with the local material control logic of topology optimization. The Energy Finite Element Method (EFEM) uses vibration energy as the basic variable and employs the finite element discretization approach to solve the energy control equations. It can achieve efficient and accurate solutions for high-frequency vibration responses under sparse meshes, balancing computational efficiency with the ability to characterize local responses, thus providing a feasible analytical basis for the optimization of high-frequency vibration structures.

[0004] Topology optimization technology can autonomously find the optimal material layout under given design domain and constraints, breaking through the limitations of traditional empirical design and serving as a core means to achieve synergistic improvement in structural lightweighting and performance. Among them, the Variable Density Method (SIMP method) has become the mainstream method for topology optimization of continuum structures due to its simple mathematical model, good numerical stability, and strong engineering adaptability, and has been widely used in static, low-frequency dynamic, and single-physics optimization scenarios. However, existing optimization methods based on the SIMP method are mostly geared towards low-frequency linear dynamics or independent thermo / mechanical single-field design, and are difficult to directly adapt to the dual special conditions of thermo-vibration coupling and high-frequency vibration.

[0005] In summary, existing technologies cannot simultaneously solve the three core problems of accurate modeling of strongly coupled thermal vibration, efficient analysis of high-frequency vibration, and adaptation to the SIMP optimization framework, making it difficult to achieve topology optimization design of thermally coupled high-frequency structures. Therefore, there is an urgent need to invent a high-frequency topology optimization method for thermally coupled structures that integrates energy finite element method and SIMP method, breaking through the technical barriers of multi-physics coupling, high-frequency dynamics, and topology optimization, and meeting the lightweight and high-performance design requirements of high-end equipment under extreme thermal vibration conditions. Summary of the Invention

[0006] To address the shortcomings of existing technologies, the present invention aims to provide a topology optimization method for mid-to-high frequency thermal vibration structures based on the variable density method. Using SIMP as the basic optimization tool and combining it with energy finite element analysis, the method guides and enhances the dissipation of vibration energy in the structure through stiffener design, thereby obtaining a more accurate high-frequency dynamic response analysis and structural design scheme, and ultimately achieving effective suppression of the high-frequency dynamic response of the structure.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A topology optimization method for mid-to-high frequency thermal vibration structures based on the variable density method includes the following steps: 1) Define the design object: The design domain is a rectangular stiffened plate, which consists of a base plate and stiffeners. The base plate dimensions are as follows: l × w × h plate The length of the substrate is l Width is w Thickness is h plate The thickness of the reinforcing rib is h stiffener The stiffening plate is fixed on all four sides, with a central load applied, and the temperature is... T The Young's modulus of the selected material at this temperature E and coefficient of thermal expansion α ; 2) Define design variables: Based on the SIMP material interpolation model, the discrete topology optimization problem is transformed into a continuous optimization problem by introducing intermediate density elements; a penalty strategy is applied to the intermediate density values ​​in the design variables to push the optimization results toward a 0-1 discrete distribution; the material interpolation model used is: (1) In the formula: E ( c i () represents the elastic modulus after interpolation; E 0 represents the elastic modulus of the material in the solid part; E min This indicates the elastic modulus of the material in the pore portion; c i For the first i The relative density of each unit, a value of 1 indicates the presence of material, and a value of 0 indicates the absence of material, i.e., voids; p As a penalty factor; In topology optimization, the SIMP method is used to discretize the design domain, dividing it into... n A 4-node rectangular element mesh, the total number of design variables is . n And assign a density design variable to each unit. c e Its value range is [0,1], and the total number of design variables is . n Design variables based on density c e As the optimization object, combined with the penalty mechanism in the SIMP method, under the condition of satisfying the total volume constraint, the design variables are updated iteratively to gradually seek the material distribution form that satisfies the objective function; 3) Finite element calculation of thermal vibration coupling energy: 3.1) Constructing the energy matrix of the thermally coupled unit: The governing differential equation for the elastic wave energy balance in each unit is: (2) In the formula: π in For input energy; e Energy density; or For damping; oh The angular frequency of the load; c g Let be the group velocity of the elastic wave. The formula for calculating the group velocity is: (3) In the formula: k Total wavenumber; D The bending stiffness of the plate; rThe density of the material; h For plate thickness; B These are terms related to thermal stress; among them k and B The solution formula is: (4) (5) In the formula: N x , N y and N xy This refers to the in-plane film force caused by thermal stress. i The direction of wave propagation; (6) In the formula: h The thickness of the plate; s x ,s y and s xy They are respectively x , y Normal and tangential thermal stress in the direction; The matrix form of the governing equations for the energy density field is: (7) in: (8) (9) (10) In the formula: K e The unit energy matrix; e e Unit energy density; F e For unit input power; Γ e For unit boundaries; Q e Energy flow at the unit boundary; The normal vector of the component boundary; N It is a shape function; 3.2) Coupled element analysis: 3.2.1) Coupling between elements of different thicknesses: When new nodes are added to the boundary of adjacent elements of different thicknesses, the energy finite element expression becomes: (11) In the formula, KThis represents the uncoupled global energy matrix. K q This is the coupling matrix between adjacent units; 3.2.2) Inter-element coupling of the same thickness: New nodes are added to the boundaries of all elements in the structure to obtain a new energy finite element mesh—an element-independent mesh. The assembly steps of the global energy matrix in the element-independent mesh are as follows: 3.2.2.1) Assembly of uncoupled element matrices with no common nodes in an independent element mesh: (12) 3.2.2.2) Recoupling rules between elements: Assuming nodes i , l , m , n Belongs to unit A, node j , k , p , q Belongs to element B; in an independent element mesh, the node i and nodes j The contributions to the uncoupled total energy matrix are respectively written as K i,i K i,l K i,m K i,n and K j,i K j,l K j,m K j,n ; will node i and nodes j The contributions in the matrix are respectively superimposed on their coupling points, that is, K i,i K i,l K i,m K i,n Superimposed on K j,i K j,l K j,m K j,n and K j,j K j,k K j,p K j,q Superimposed on K i,j K i,k K i,p K i,q ; 3.2.2.3) Couple the horizontal and vertical boundary element nodes according to the rules in step 3.2.2.2); 3.2.2.4) Couple the diagonal boundary element nodes according to the rules in step 3.2.2.2); 3.3) Determine the optimization model: With the minimum structural energy flexibility as the optimization objective, the material usage is used as the constraint function. During implementation, the objective function and constraint function are determined based on actual needs. The optimized mathematical model is as follows: (13) In the formula: c For design variables, where c i For the first i The relative density of each unit n e The total number of units; M ( c ) is the constraint function. V ( c )and V 0 represents the actual material usage and the initial material usage in the design domain, respectively. f The target volume fraction; J ( c ) represents the structural energy compliance; 3.4) Sensitivity analysis: Calculate the sensitivity of the objective function and constraint functions to the design variables; 4) Iterative optimization: The calculation results and sensitivity information obtained from the energy finite element analysis are input into the moving asymptote algorithm (MMA). The design variables are updated iteratively to make the objective function gradually converge under the given constraints, and finally the optimal topology of the stiffened plate that meets the material usage limit is obtained. 5) Adaptive processing: The topology of the stiffened plate is adapted and rounded according to the production process requirements to obtain the final structure layout that can be manufactured.

[0008] Compared with existing technologies, the beneficial effects of this invention are as follows: Because this invention adopts a thermo-vibration coupling modeling approach, abandoning the decoupling modeling approach of existing technologies, it incorporates the modulation effect of the temperature field on structural material properties and stiffness, along with the feedback influence of high-frequency vibration energy transfer, into a unified model. Simultaneously, it uses the energy finite element method to solve the high-frequency vibration response, thus possessing the advantages of strong adaptability to thermo-vibration coupling conditions and high computational efficiency for high-frequency response. This solves the technical problems of high high-frequency computation cost and the inability of statistical energy analysis to characterize local details in traditional finite element methods. Furthermore, because this invention deeply integrates energy finite element high-frequency analysis with the SIMP method, it constructs a thermo-vibration coupling topology optimization framework adapted to high-frequency conditions, thus possessing the advantages of fast iterative convergence speed and good numerical stability, effectively avoiding the problem of large deviations between optimization results and actual operating conditions in existing technologies. Finally, because this invention achieves multi-objective collaborative optimization for lightweighting and vibration reduction, rather than single-field or low-frequency optimization in existing technologies, it can provide optimal topology configurations that better fit actual harsh operating conditions for key components in aerospace, high-end equipment, and other fields, significantly improving structural reliability and service life, resulting in higher engineering application value and a wider range of applications. Attached Figure Description

[0009] Figure 1 This is a schematic diagram of boundary conditions in an embodiment of the present invention.

[0010] Figure 2 This is a schematic diagram of an independent grid unit according to an embodiment of the present invention.

[0011] Figure 3 This is a schematic diagram illustrating the definition of an independent grid node in an embodiment of the present invention.

[0012] Figure 4 This is a diagram showing the final optimized result of an embodiment of the present invention.

[0013] Figure 5 This is a comparative structural diagram of an embodiment of the present invention. Detailed Implementation

[0014] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0015] A topology optimization method for mid-to-high frequency thermal vibration structures based on the variable density method includes the following steps: 1) Define the design object: Use the rectangular stiffened slab as the design domain, such as... Figure 1 As shown, the stiffening plate consists of a base plate and stiffeners. The base plate has dimensions of 500mm × 500mm × 5mm, and the stiffeners are 5mm thick. The stiffening plate is fixed on all four sides. A 4000Hz, 1000W energy input is applied to the center of the stiffening plate. The temperature of the structure is... T The temperature is 260℃, and the Young's modulus of the selected material at that temperature is... E =56.3 GPa and coefficient of thermal expansionα =2.67×10 -5 K -1 ; 2) Define design variables: Based on the SIMP material interpolation model, the discrete topology optimization problem is transformed into a continuous optimization problem by introducing intermediate density elements. Intermediate density elements are physically difficult to realize and cannot be manufactured; therefore, their generation should be avoided as much as possible, and their number should be reduced. To achieve this goal, a penalty strategy needs to be applied to the intermediate density values ​​in the design variables to push the optimization results towards a 0-1 discrete distribution. The material interpolation model used is: (1) In the formula: E ( c i () represents the elastic modulus after interpolation; E 0 represents the elastic modulus of the solid part of the material. E 0 = 71 GPa; E min This represents the elastic modulus of the material in the pore portion. E min =0.001 E 0; c i For the first i The relative density of each unit, a value of 1 indicates the presence of material, and a value of 0 indicates the absence of material, i.e., voids; p As a penalty factor, p =3; In topology optimization, the aforementioned SIMP method is used to discretize the design domain, dividing it into... n A 4-node rectangular element mesh, the total number of design variables is . n And assign a density design variable to each unit. c e Its value range is [0,1], and the total number of design variables is . n Design variables based on these densities c e As the optimization object, combined with the penalty mechanism in the SIMP method, under the condition of satisfying the total volume constraint, the design variables are updated iteratively to gradually seek the material distribution form that satisfies the objective function; 3) Finite element calculation of thermal vibration coupling energy: 3.1) Constructing the energy matrix of the thermally coupled unit: The governing differential equation for the elastic wave energy balance in each unit is: (2) In the formula: π in For input energy;e Energy density; or For damping; oh The angular frequency of the load; c g Let be the group velocity of the elastic wave. The formula for calculating the group velocity is: (3) In the formula: k Total wavenumber; D The bending stiffness of the plate; r The density of the material; h For plate thickness; B These are terms related to thermal stress; among them k and B The solution formula is: (4) (5) In the formula: N x , N y and N xy This refers to the in-plane film force caused by thermal stress. i The direction of wave propagation; (6) In the formula: h The thickness of the plate; s x ,s y and s xy They are respectively x , y Normal and tangential thermal stress in the direction; The matrix form of the energy density field governing equations is as follows: (7) in: (8) (9) (10) In the formula: K e The unit energy matrix; e e Unit energy density; F e For unit input power; Γ e For unit boundaries; Q e Energy flow at the unit boundary; The normal vector of the component boundary; N It is a shape function; 3.2) Coupled element analysis: 3.2.1) Coupling between elements of different thicknesses: Adding new nodes (e.g., on the boundary of adjacent elements of different thicknesses) Figure 2 As shown), the finite element expression for energy is: (11) In the formula, K This represents the uncoupled global energy matrix. K q This is the coupling matrix between adjacent units; 3.2.2) Inter-element coupling of the same thickness: Adding new nodes to the boundaries of all elements in the structure yields a new energy finite element mesh—an element-independent mesh (e.g., Figure 3 As shown), the steps for assembling the global energy matrix in an independent cell mesh are as follows: 3.2.2.1) Assembly of uncoupled element matrices with no common nodes in an independent element mesh: (12) 3.2.2.2) Recoupling rules between elements: Assuming nodes i , l , m , n Belongs to unit A, node j , k , p , q Belongs to unit B, in Figure 3 In the cell-independent mesh shown, nodes i and nodes j The contributions to the uncoupled total energy matrix are respectively written as K i,i K i,l K i,m K i,n and K j,i K j,l K j,m K j,n ; will node i and nodes j The contributions in the matrix are respectively superimposed on their coupling points, that is, K i,i K i,l K i,m K i,n Superimposed on K j,i K j,l K j,m K j,n and K j,j K j,k Kj,p K j,q Superimposed on K i,j K i,k K i,p K i,q ; 3.2.2.3) Coupling of horizontal and vertical boundary element nodes: such as Figure 2 As shown, to achieve coupling between elements 3 & 4 and elements 2 & 4, nodes 6 & 13, 7 & 16, 11 & 14, and 12 & 13 need to be recoupled respectively; to achieve recoupling of node 6 & 13, K should be... 6,5 K 6,6 K 6,7 K 6,8 Stack it onto columns 5, 6, 7, and 8 of row 13, then add K. 13,13 K 13,14 K 13,15 K 13,16 Overlay them onto columns 13, 14, 15, and 16 in row 6; perform the same operation on nodes 7 & 16, 11 & 14, and 12 & 13 to achieve recoupling; 3.2.2.4) Coupling of diagonal boundary element nodes; such as Figure 2 As shown, diagonal nodes 6 & 12 are coupled together, and K is... 6,5 K 6,6 K 6,7 K 6,8 Stack it onto columns 5, 6, 7, and 8 of row 12, then add K. 12,9 K 12,10 K 12,11 K 12,12 Stack it onto columns 9, 10, 11, and 12 in row 6; (13) 3.2.3) The final form of the global energy matrix: (14) 3.3) Determine the optimization model: In this embodiment, the energy compliance of the stiffened plate is used. J ( c The objective function is defined as follows: the amount of reinforcing rib material used must not exceed 30% of the amount of substrate material used. M ( c )≤ f =30%; The optimized mathematical model is as follows: (15) In the formula: c For design variables, where ci For the first i The relative density of each unit, the total number of units n e It is 1600; M ( c ) is the constraint function. V ( c (This represents the actual amount of material used.) V 0 represents the initial material usage for the design domain. V 0 = 0.5 f For the target volume fraction, f =0.3; J ( c ) represents the structural energy compliance; 3.4) Sensitivity Analysis: 3.4.1) Sensitivity of the objective function: (16) 3.4.2) Sensitivity of constraint functions: (17) In the formula: v e Representation unit e volume, V 0 represents the initial material usage for the design domain; 4) Iterative optimization: The calculation results and sensitivity information obtained from the energy finite element analysis are input into the moving asymptote algorithm (MMA). The design variables are updated iteratively to make the objective function gradually converge under the given constraints, and finally the optimal topology of the stiffened plate that meets the material usage limit is obtained. 5) Adaptive processing: The topology of the stiffened plate is adapted and rounded according to the production process requirements to obtain a final manufacturable structural layout. For example... Figure 4 , Figure 5 As shown, it can be seen Figure 4 It exhibits a centrally symmetrical, cross-shaped topological structure, with material distribution concentrated in the central region and extending branches in all four directions. The overall form is compact and possesses distinct mechanical transmission path characteristics. Figure 5 It is a regular, grid-like traditional structure with uniform material distribution and periodic repetition. In comparison, Figure 4 The structure is more directional in energy transfer and stress distribution, and can respond more efficiently to the thermal coupling requirements under high-frequency vibration. Figure 5While regular meshes possess uniform load-bearing capacity, their performance in targeted vibration suppression and energy dissipation is relatively weak. This embodiment employs thermal-vibration coupling modeling combined with the energy finite element method to solve high-frequency responses, efficiently adapting to operating conditions and addressing the pain points of traditional methods. It integrates the energy finite element method with the SIMP method to construct a high-frequency coupling optimization framework, exhibiting fast convergence and good stability. This achieves synergistic optimization of lightweighting and vibration reduction, providing optimal topologies for critical components in aerospace and high-end equipment that meet stringent operating conditions, improving structural reliability and service life. It has high engineering application value and a wide range of applicability.

Claims

1. A topology optimization method for medium- and high-frequency thermally vibrating structures based on the variable density method, characterized in that, Includes the following steps: 1) Define the design object: The design domain is a rectangular stiffened plate, which consists of a base plate and stiffeners. The base plate dimensions are as follows: l × w × h plate The length of the substrate is l Width is w Thickness is h plate The thickness of the reinforcing rib is h stiffener The stiffening plate is fixed on all four sides, with a central load applied, and the temperature is... T The Young's modulus of the selected material at this temperature E and coefficient of thermal expansion α ; 2) Define design variables: Based on the SIMP material interpolation model, the discrete topology optimization problem is transformed into a continuous optimization problem by introducing intermediate density elements; a penalty strategy is applied to the intermediate density values ​​in the design variables to push the optimization results toward a 0-1 discrete distribution; the material interpolation model used is: （1） In the formula: E ( γ i () represents the elastic modulus after interpolation; E 0 represents the elastic modulus of the material in the solid part; E min This indicates the elastic modulus of the material in the pore portion; γ i For the first i The relative density of each unit, a value of 1 indicates the presence of material, and a value of 0 indicates the absence of material, i.e., voids; p As a penalty factor; 3) Finite element calculation of thermal vibration coupling energy: 3.1) Constructing the energy matrix of the thermally coupled unit: 3.2) Coupled element analysis: 3.3) Determine the optimization model: With the minimum structural energy flexibility as the optimization objective, the material usage is used as the constraint function. During implementation, the objective function and constraint function are determined based on actual needs. 3.4) Sensitivity analysis: Calculate the sensitivity of the objective function and constraint functions to the design variables; 4) Iterative optimization: The calculation results and sensitivity information obtained from the energy finite element analysis are input into the moving asymptote algorithm (MMA). The design variables are updated iteratively to make the objective function gradually converge under the given constraints, and finally the optimal topology of the stiffened plate that meets the material usage limit is obtained. 5) Adaptive processing: The topology of the stiffened plate is adapted and rounded according to the production process requirements to obtain the final structure layout that can be manufactured.

2. The method for topology optimization of medium- and high-frequency thermally vibrating structures based on the variable density method according to claim 1, characterized in that, 3.1) The specific steps for constructing the energy matrix of the thermally coupled unit are as follows: The governing differential equation for the elastic wave energy balance in each unit is: （2） In the formula: π in For input energy; e Energy density; η For damping; ω The angular frequency of the load; c g Let be the group velocity of the elastic wave. The formula for calculating the group velocity is: （3） In the formula: k Total wavenumber; D The bending stiffness of the plate; ρ The density of the material; h For plate thickness; B These are terms related to thermal stress; among them k and B The solution formula is: （4） （5） In the formula: N x , N y and N xy This refers to the in-plane film force caused by thermal stress. θ The direction of wave propagation; （6） In the formula: h The thickness of the plate; σ x σ y and σ xy They are respectively x , y Normal and tangential thermal stress in the direction; The matrix form of the energy density field governing equations is as follows: （7） in: （8） （9） （10） In the formula: K e The unit energy matrix; e e Unit energy density; F e For unit input power; Γ e For unit boundaries; Q e Energy flow at the unit boundary; The normal vector of the component boundary; N It is a shape function.

3. The topology optimization method for medium- and high-frequency thermal vibration structures based on the variable density method according to claim 1, characterized in that step 3.2) of the coupling element analysis specifically comprises: 3.2.1) Coupling between elements of different thicknesses: When new nodes are added to the boundary of adjacent elements of different thicknesses, the energy finite element expression becomes: （11） In the formula, K This represents the uncoupled global energy matrix. K q This is the coupling matrix between adjacent units; 3.2.2) Inter-element coupling of the same thickness: New nodes are added to the boundaries of all elements in the structure to obtain a new energy finite element mesh—an element-independent mesh. The assembly steps of the global energy matrix in the element-independent mesh are as follows: 3.2.2.1) Assembly of uncoupled element matrices with no common nodes in an independent element mesh: （12） 3.2.2.2) Recoupling rules between elements: Assuming nodes i , l , m , n Belongs to unit A, node j , k , p , q Belongs to element B; in an independent element mesh, nodes i and nodes j The contributions to the uncoupled total energy matrix are respectively written as K i,i K i,l K i,m K i,n and K j,i K j,l K j,m K j,n ; will node i and nodes j The contributions in the matrix are respectively superimposed on their coupling points, that is, K i,i K i,l K i,m K i,n Superimposed on K j,i K j,l K j,m K j,n and K j,j K j,k K j,p K j,q Superimposed on K i,j K i,k K i,p K i,q ; 3.2.2.3) Couple the horizontal and vertical boundary element nodes according to the rules in step 3.2.2.2); 3.2.2.4) Couple the diagonal boundary element nodes according to the rules in step 3.2.2.2).

4. The method for topology optimization of medium- and high-frequency thermally vibrating structures based on the variable density method according to claim 1, characterized in that, Step 3.3) Determine the optimization mathematical model in the optimization model as follows: （13） In the formula: γ For design variables, where γ i For the first i The relative density of each unit n e The total number of units; M ( γ ) is the constraint function. V ( γ )and V 0 represents the actual material usage and the initial material usage in the design domain, respectively. f The target volume fraction; J ( γ ) represents the structural energy flexibility.

5. The method for topology optimization of medium- and high-frequency thermally vibrating structures based on the variable density method according to claim 1, characterized in that, Step 2) In topology optimization, the SIMP method is used to discretize the design domain, dividing it into... n A 4-node rectangular element mesh, the total number of design variables is . n And assign a density design variable to each unit. γ e Its value range is [0,1], and the total number of design variables is . n ; Design variables based on density γ e As the optimization target, combined with the penalty mechanism in the SIMP method, under the condition of satisfying the total volume constraint, the design variables are updated iteratively to gradually seek the material distribution form that satisfies the objective function.

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