Medium-high frequency thermal vibration coupling structure topological optimization design method based on energy finite element

Abstract

A medium-high frequency thermal vibration coupling structure topology optimization design method based on an energy finite element takes an explicit topology optimization method as a tool, combines an energy finite element analysis method, obtains more accurate dynamic response analysis and structure design results through reinforcing rib design guiding and strengthening structure vibration energy dissipation, and achieves suppression of structure dynamic response. According to the method, the problems of low calculation precision, large calculation amount and the like in the optimization design of the large-scale high-frequency vibration structure are solved; and meanwhile, the method can be further popularized to applications such as curved surfaces, space structures and composite materials by modifying corresponding energy finite element modules.

Description

Technical Field

[0001] This invention belongs to the field of dynamic performance optimization design technology of structures under thermal conditions, and specifically relates to a topology optimization design method for medium and high frequency thermal vibration coupled structures based on energy finite element method. Background Technology

[0002] With the continuous advancement of advanced equipment, equipment is developing towards higher precision and lighter weight. Most high-end equipment is subjected to increasingly severe thermal environments and mid-to-high frequency vibrations during service. High-frequency vibrations under harsh thermal environments cause unnecessary fatigue and noise in the structure, directly affecting its performance, structural safety, and reliability. To ensure the performance and safety of equipment, it is essential to pay close attention to the mid-to-high frequency response characteristics of structures under thermal environments. However, the mid-to-high frequency dynamic optimization of structures under thermal environments faces two key challenges: the calculation methods for dynamic response and the methods for optimization design. Therefore, the control of high-frequency vibrations in structures under thermal environments presents a challenge.

[0003] When calculating the high-frequency response of structures using the Finite Element Method (FEA), as exemplified by the patent application titled "A Method for Predicting High-Frequency Acoustic Radiation of Shells Based on Acousto-Structure Coupling and Finite Element Analysis" (Publication No. CN118296911A), increasingly dense finite element meshes are required as the calculation frequency increases. This is particularly problematic for large-scale structures, where the total number of elements increases dramatically, significantly impacting computational efficiency and even rendering the method impractical. The Statistical Energy Analysis (SEA) method can only respond to the average energy level of substructures, as seen in the patent application titled "A Method for Predicting High-Frequency Environment of Bundled Rockets Based on Statistical Energy Analysis" (Publication No. CN113642097A). It cannot accurately describe the energy response information of individual material points within the substructure. Therefore, SEA-based optimization designs typically employ dimensional optimization, using thickness and material properties as design variables, making it difficult to achieve local topological changes within the subsystem.

[0004] Topology optimization is widely used due to its greater design freedom and larger design space. However, because the topology optimization process requires iterative calculations and the energy response of each material point in the structure, classical finite element method and statistical energy method are not the preferred computational tools for topology optimization of high-frequency vibration structures under thermal conditions. Summary of the Invention

[0005] To overcome the shortcomings of the prior art, the present invention aims to provide a topology optimization design method for medium- and high-frequency thermal vibration coupled structures based on energy finite element method. Using explicit topology optimization method as a tool, combined with energy finite element analysis method, the method guides and enhances the dissipation of structural vibration energy through stiffener design, thereby obtaining more accurate dynamic response analysis and structural design results and achieving suppression of structural dynamic response.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A topology optimization design method for mid-to-high frequency thermal-vibration coupled structures based on energy finite element method includes the following steps: 1) Define the design object: The design domain is the stiffened plate, which consists of a base plate and stiffeners. The base plate dimensions are as follows: l × w × h plate , l For the length of the substrate, w The width of the substrate, h plate The thickness of the substrate is [value], and the thickness of the reinforcing rib is [value]. h stiffener The stiffening plate is fixed at its four corners, and the center of the stiffening plate is the energy load input area, where the temperature is... t And the Young's modulus of the selected material at that temperature. E and coefficient of thermal expansion α ; 2) Define design variables: Based on the explicit level set description method, rectangular components are used to describe the stiffeners. Each component contains center coordinates ( x 0, y 0) Length L, width T, and tilt angle f A total of 5 variables are evenly distributed on the substrate. n These components are used as the initial layout for reinforcing ribs on the substrate, resulting in a total of 5 components. n One design variable; 3) Explicit geometric description of stiffeners: 3.1) Constructing the first using geometric parameters q The level set function of each component : (1) in: (2) In the formula: These are local coordinates (coordinates after rotation and translation relative to the component's center coordinates). p To control the shape index; x , y () represents the coordinates of the element node; 3.2) Using the Heaviside function to apply the level set function Perform two normalizations to obtain the horizontal lumped set. : (3) In the formula: For the Heaviside function; bIt is a large positive number used to sharpen the boundaries. b The larger the value, the closer the merged interface is to the ideal Heaviside transition; n com The total number of components; 4) Projecting the topological description of the structure onto the energy finite element model: The element thickness within the energy finite element mesh is determined by the coverage of the component (i.e., stiffener) on that element. (4) In the formula: h j For unit j The equivalent thickness; h stiffener and h plate The thicknesses of the reinforcing ribs and the substrate are predefined, respectively. For unit j Middle node i The level set value; n nod The total number of nodes involved in each unit; 5) Finite element calculation of thermal vibration coupling energy: 5.1) Constructing the energy matrix of the thermally coupled unit: The governing differential equation for the elastic wave energy balance in each unit is: (5) In the formula: π in To 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: (6) 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: (7) (8) In the formula N x , N y and N xyIn-plane film force caused by thermal stress, i The direction of wave propagation; (9) 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: (10) in: (11) (12) (13) 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; 5.2) Coupled element analysis: 5.2.1) 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: (14) In the above formula, K This represents the uncoupled global energy matrix. K q This is the coupling matrix between adjacent units; 5.2.2) 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: 5.2.2.1) Assembly of uncoupled element matrices with no common nodes in an independent element mesh: (15) 5.2.2.2) Recoupling rules between elements: Assuming nodesi , l , m , n Belongs to unit A, node j , k , p , q Belonging to element B, in an element-independent mesh, 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 ; 5.2.2.3) Couple the vertical boundary element nodes according to the rules in step 5.2.2.2); 5.2.2.4) Couple the diagonal boundary element nodes according to the rules in step 5.2.2.2); 5.3) Determine the optimal mathematical model: With the optimal dynamic performance of the thermally vibrating structure as the optimization objective, material usage, stress, and manufacturing constraints are used as constraint functions. During implementation, the objective function and constraint functions are determined based on actual needs. The optimized mathematical model is as follows: (16) In the formula: c A vector storing the geometric characteristic variables of the stiffeners, where c j Including reinforcing ribs j length L j ,width T j Inclination angle fj and the coordinates of the center point; M ( c ) represents the constraint function; J ( c Let ) be the objective function; 5.4) Sensitivity Analysis: Calculate the sensitivity of the objective function and constraint functions to the design variables; 6) Iterative optimization: Substitute the energy finite element calculation results and sensitivity into the moving asymptote optimization algorithm, iteratively update the variables until the objective function converges under the constraint conditions. At this point, the optimal structural layout of the stiffened plate under the material usage constraint conditions is obtained. 7) Adaptive processing: The optimal structural layout of the stiffened plate is rounded according to the production process requirements, so as to obtain the final structural layout of the stiffened plate.

[0007] Compared with the prior art, the beneficial effects of the present invention are: Because this invention adopts an innovative framework combining energy finite element analysis and explicit topology optimization, it abandons the dependence of the classical finite element method (FEA) on dense meshes in high-frequency calculations and overcomes the limitation of the statistical energy method (SEA) in accurately describing local energy responses. Therefore, it has the advantages of high computational efficiency and accuracy, effectively solving the core problem of large computational load in the optimization of large-scale high-frequency vibration structures. It can efficiently obtain energy response information of each material point in the structure, providing reliable data support for topology optimization. Furthermore, the layout of stiffeners can be flexibly adjusted during the optimization process to guide vibration energy dissipation. Compared with traditional dimensional optimization, it can explore a wider design space and achieve in-depth optimization of the structure's dynamic performance. This invention is applicable to the high-frequency vibration topology optimization framework of structures under thermal environments; it can also be further extended to applications such as curved surfaces, spatial structures, and composite materials by modifying the corresponding EFEM module. Attached Figure Description

[0008] Figure 1 This is a schematic diagram of boundary conditions in an embodiment of the present invention.

[0009] Figure 2 This is the initial layout diagram of the components in this embodiment of the invention.

[0010] Figure 3 This is a schematic diagram of an independent grid unit according to an embodiment of the present invention.

[0011] Figure 4 This is a schematic diagram illustrating the definition of an independent grid node in an embodiment of the present invention.

[0012] Figure 5 This is the final optimization result that satisfies the constraints in the embodiments of the present invention. Detailed Implementation

[0013] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0014] A topology optimization design method for mid-to-high frequency thermal-vibration coupled structures based on energy finite element 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 4000mm × 4000mm × 50mm, and the stiffeners are 50mm thick. The four corners of the stiffening plate are fixed, and the center of the stiffening plate receives an input of 1000Hz, 8000W of energy. The structure is located at a temperature of 260℃, and the Young's modulus of the selected material at this temperature is also specified. E =56.3 GPa and coefficient of thermal expansion α =2.67×10 -5 K -1 ; 2) Define design variables: Based on the explicit level set description method, rectangular components are used to describe the stiffeners. Each component contains center coordinates ( x 0, y 0) Length L, width T, and tilt angle f A total of 5 variables were used, and 32 components were evenly arranged on the substrate as the initial layout for the reinforcing ribs on the substrate, such as... Figure 2 As shown, there are currently 160 design variables. 3) Explicit geometric description of stiffeners: 3.1) Constructing the first using geometric parameters q The level set function of each component : (1) in: (2) In the formula: These are local coordinates (coordinates after rotation and translation relative to the component's center coordinates). p To control the shape index, at this time p =6;( x , y () represents the coordinates of the element node; 3.2) Using the Heaviside function to apply the level set function Perform two normalizations to obtain the horizontal lumped set. : (3) In the formula: For the Heaviside function; b It is a large positive number used to sharpen the boundaries. bThe larger the value, the closer the merged interface is to the ideal Heaviside transition. b =10000; n com This represents the total number of components. n com =32; 4) Projecting the topological description of the structure onto the energy finite element model: The element thickness within the energy finite element mesh is determined by the coverage of the component (i.e., stiffener) on that element. (4) In the formula: h j For unit j The equivalent thickness; h stiffener and h plate The thicknesses of the reinforcing ribs and the substrate are predefined, respectively. For unit j Middle node i The level set value; n nod The total number of nodes involved in each unit; 5) Finite element calculation of thermal vibration coupling energy: 5.1) Constructing the energy matrix of the thermally coupled unit: The governing differential equation for the elastic wave energy balance in each unit is: (5) In the formula: π in To 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: (6) 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: (7) (8) In the formula N x , N y andN xy In-plane film force caused by thermal stress, i The direction of wave propagation; (9) 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: (10) in: (11) (12) (13) 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; 5.2) Coupled element analysis: 5.2.1) Coupling between elements of different thicknesses: Adding new nodes (e.g., on the boundary of adjacent elements of different thicknesses) Figure 3 As shown), the finite element expression for energy is: (14) In the above formula, K This represents the uncoupled global energy matrix. K q This is the coupling matrix between adjacent units; 5.2.2) 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: 5.2.2.1) Assembly of uncoupled element matrices with no common nodes in an independent element mesh: (15) 5.2.2.2) 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 4 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 K j,p K j,q Superimposed on K i,j K i,k K i,p K i,q ; 5.2.2.3) Coupling of horizontal and vertical boundary element nodes: such as Figure 3 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 13,16Overlay 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; 5.2.2.4) Coupling of diagonal boundary element nodes; such as Figure 3 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; (16) 5.2.3) The final form of the global energy matrix: (17) 5.3) Determine the optimal mathematical model: In this embodiment, the energy compliance of the stiffened plate is used. J ( c The objective function is ; the constraint is that the amount of reinforcing rib material used must not exceed 30% of the amount of substrate material used. M ( c )≤ M upp =30%; The optimized mathematical model is as follows: (18) In the formula: c A vector storing the geometric characteristic variables of the stiffeners, where c j Including reinforcing ribs j length L j ,width T j Inclination angle f j and the coordinates of the center point; M ( c )and M upp These are the actual material usage and the upper limit of material usage, respectively. J ( c () represents the structural energy flexibility; the total number of elements n e It is 1600. h j and S j These are the thickness of the reinforcing ribs and the reinforcing ribs themselves.j Top surface area; 5.4) Sensitivity Analysis: 5.4.1) The formula for calculating the sensitivity of the objective function is as follows: (19) 5.4.2) The formula for calculating the sensitivity of the constraint function is as follows: (20) In the formula: n e The total number of units; n nod The total number of nodes; h stiffener For the thickness of the reinforcing ribs; For horizontal integration; 6) Iterative optimization: Substitute the energy finite element calculation results and sensitivity into the moving asymptote optimization algorithm, iteratively update the variables until the objective function converges under the constraint conditions. At this point, the optimal structural layout of the stiffened plate under the material usage constraint conditions is obtained. 7) Adaptive processing: The optimal structural layout of the stiffened plate is rounded according to the production process requirements to obtain the final structural layout of the stiffened plate, such as... Figure 5 As shown, comparison Figure 2 It can be seen that the irregular, multi-branch "quasar" topology layout optimized by the method of this invention is, compared to Figure 2 The traditional grid-like configuration is based on the result of precise optimization of energy response. Because this invention employs a framework combining energy finite element method and explicit topology optimization, this configuration can more efficiently guide the dispersion and dissipation of vibration energy, avoiding the local energy concentration problem that easily occurs in traditional grid configurations. Simultaneously, the topology achieved through the level set method better matches the performance requirements under thermal-vibration coupling conditions in terms of material distribution, reducing unnecessary material redundancy and strengthening the structural stiffness of key areas. Compared to traditional configurations, it can better achieve vibration suppression with the same amount of material, making it a more efficient and optimized configuration better suited to actual service conditions.

[0015] This invention proposes a computationally efficient and applicable high-frequency vibration topology optimization framework for structures under thermal conditions, based on the energy finite element method and utilizing an explicit level set description method. Furthermore, by modifying the corresponding EFEM module, it can be further extended to applications such as curved surfaces, spatial structures, and composite materials.

Claims

1. A topology optimization design method for mid-to-high frequency thermal-vibration coupled structures based on energy finite element method, characterized in that, Includes the following steps: 1) Define the design object: The design domain is the stiffened plate, which consists of a base plate and stiffeners. The base plate dimensions are as follows: l × w × h plate , l The length of the substrate, w The width of the substrate. h plate The thickness of the substrate is [value]; the thickness of the reinforcing rib is [value]. h stiffener The stiffening plate is fixed at its four corners, and the center of the stiffening plate is the energy load input area, where the temperature is... t And the Young's modulus of the selected material at that temperature. E and coefficient of thermal expansion α ; 2) Define design variables: Based on the explicit level set description method, rectangular components are used to describe the stiffeners. Each component contains center coordinates ( x 0, y 0) Length L, width T, and tilt angle φ A total of 5 variables are evenly distributed on the substrate. n These components are used as the initial layout for reinforcing ribs on the substrate, resulting in a total of 5 components. n One design variable; 3) Explicit geometric description of stiffeners: 3.1) Constructing the first using geometric parameters q The level set function of each component : （1） in: （2） In the formula: These are local coordinates, meaning the coordinates after rotation and translation relative to the component's center coordinates. p To control the shape index; x , y () represents the coordinates of the element node; 3.2) Using the Heaviside function to apply the level set function Perform two normalizations to obtain the horizontal lumped set. : （3） In the formula: For the Heaviside function; b It is a large positive number used to sharpen the boundaries. b The larger the value, the closer the merged interface is to the ideal Heaviside transition; n com The total number of components; 4) Projecting the topological description of the structure onto the energy finite element model: The element thickness within the energy finite element mesh is determined by the coverage of the component, i.e., the stiffener, over that element. （4） In the formula: h j For unit j The equivalent thickness; h stiffener and h plate The thicknesses of the reinforcing ribs and the substrate are predefined, respectively. For unit j Middle node i The level set value; n nod The total number of nodes involved in each unit; 5) Finite element calculation of thermal vibration coupling energy: 5.1) Constructing the energy matrix of the thermally coupled unit: 5.2) Coupled element analysis: 5.3) Determine the optimal mathematical model: With the optimal dynamic performance of the thermally vibrating structure as the optimization objective, material usage, stress, and manufacturing constraints are used as constraint functions. During implementation, the objective function and constraint functions are determined based on actual needs. 5.4) Sensitivity Analysis: Calculate the sensitivity of the objective function and constraint functions to the design variables; 6) Iterative optimization: Substitute the energy finite element calculation results and sensitivity into the moving asymptote optimization algorithm, iteratively update the variables until the objective function converges under the constraint conditions. At this point, the optimal structural layout of the stiffened plate under the material usage constraint conditions is obtained. 7) Adaptive processing: The optimal structural layout of the stiffened plate is rounded according to the production process requirements, so as to obtain the final structural layout of the stiffened plate.

2. The topology optimization design method for a medium-to-high frequency thermal vibration coupled structure based on energy finite element method according to claim 1, characterized in that, Step 5.1) Specifically, the governing differential equation for the elastic wave energy balance in each unit is: （5） 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: （6） 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 as follows （7） （8） 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; （9） 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 governing equations for the energy density field is: （10） in: （11） （12） （13） 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 design method for a medium-to-high frequency thermal vibration coupling structure based on energy finite element method according to claim 2, characterized in that, Step 5.2) specifically involves: 5.2.1) 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: （14） In the above formula, K This represents the uncoupled global energy matrix. K q This is the coupling matrix between adjacent units; 5.2.2) 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: 5.2.2.1) Assembly of uncoupled element matrices with no common nodes in an independent element mesh: （15） 5.2.2.2) Recoupling rules between elements: Assuming nodes i , l , m , n Belongs to unit A, node j , k , p , q Belonging to element B, in an element-independent mesh, 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 ; 5.2.2.3) Couple the vertical boundary element nodes according to the rules in step 5.2.2.2); 5.2.2.4) Couple the diagonal boundary element nodes according to the rules in step 5.2.2.2).

4. The topology optimization design method for a medium-to-high frequency thermal vibration coupling structure based on energy finite element method according to claim 3, characterized in that, The optimized mathematical model for step 5.3) is as follows: （16） In the formula: γ A vector storing the geometric characteristic variables of the stiffeners, where γ j Including reinforcing ribs j length L j ,width T j Inclination angle φ j and the coordinates of the center point; M ( γ ) represents the constraint function; J ( γ ) is the objective function.

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