A topological optimization method and system for precisely and efficiently regulating structural deformation characteristics

Abstract

This invention belongs to the field of structural optimization design, and specifically discloses a topology optimization method and system for accurately and efficiently controlling structural deformation characteristics. The method includes: performing finite element mesh generation on the structure to be optimized, which exhibits spring deformation characteristics, and defining design variables corresponding to the mesh; determining the total strain energy constraint of the structure based on the spring constant and maximum deformation, and establishing a first topology optimization model with the goal of minimizing the total structural volume; performing topology optimization based on the first topology optimization model to obtain the structural configuration and its total volume; setting an upper limit for the total structural volume constraint based on this total volume to construct the total structural volume constraint, while simultaneously using the total strain energy constraint of the first topology optimization model, and establishing a second topology optimization model with the goal of minimizing the p-norm of the local deformation error under external force; and performing topology optimization based on the second topology optimization model to obtain the final structural configuration. This invention is highly versatile, functionally diverse, and precise and efficient.

Description

Technical Field

[0001] This invention belongs to the field of structural optimization design, and more specifically, relates to a topology optimization method and system for accurately and efficiently controlling structural deformation characteristics. Background Technology

[0002] Topology optimization, a cutting-edge structural design method based on advanced mathematical algorithms, aims to explore the optimal distribution of materials in space to maximize structural properties such as stiffness and frequency, or minimize properties such as deformation and mass. This method has been widely applied in various fields such as aerospace, automotive manufacturing, biomedicine, and consumer electronics, significantly improving product performance while effectively reducing weight and cost, demonstrating substantial application advantages.

[0003] In the design of flexible mechanisms, topology optimization can generate configurations with maximum output displacement and effectively balance flexibility and stiffness requirements in maximum displacement constraints or minimization problems. However, for structural optimization design problems involving spring deformation characteristics, such as leaf springs, reeds, and integrated elastic support devices, it is difficult to directly apply the above methods. Furthermore, current research is largely limited to traditional parametric optimization methods, with little in-depth exploration of innovative topology configuration design. Therefore, research on topology optimization methods for controlling spring deformation characteristics is of great significance for overcoming the limitations of empirical design, exploring lightweight and high-performance structural forms, and promoting the integrated design of flexible mechanisms and elastic elements. It can fill the gap in theoretical and applied research in this field and has significant engineering application value for improving the performance of precision instruments, buffer devices, and other applications.

[0004] In the field of research on controlling structural deformation characteristics, most existing methods have significant shortcomings. Previous studies have largely focused on topology optimization design of flexible mechanisms, aiming to generate configurations with maximum output displacement. However, they struggle to precisely control the ratio of external force to maximum deformation in a specific region (i.e., the spring stiffness coefficient). Furthermore, the location of the external force and the output displacement are typically distributed on both sides of the structure, limiting their application in structures with spring-like deformation characteristics. In addition, a few studies have disclosed topology optimization methods oriented towards constraining or minimizing maximum displacement, but these typically only control the maximum deformation (displacement) of the structure to not exceed a specific value. They also struggle to precisely control the ratio of external force to maximum deformation in a specific region and cannot solve the problem of uniform distribution of maximum deformation. Moreover, these methods use the p-norm of displacements of all mesh elements or nodes within the design domain to characterize the maximum structural deformation. Calculating the sensitivity of each node displacement requires inverting the overall stiffness matrix, resulting in a huge computational burden and problems such as slow convergence and unclear results. For traditional helical springs with fixed shapes, parameter optimization and other methods are sufficient to improve performance. However, for structures such as leaf springs, reeds, and integrated elastic support devices, parameter optimization technology has significant limitations, and there is currently no topology optimization method for such problems.

[0005] Therefore, there is an urgent need to develop a topology optimization method that can accurately and efficiently control the deformation characteristics of springs, so as to break through the limitations of traditional design methods and obtain novel topological configurations with better performance. Summary of the Invention

[0006] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a topology optimization method and system for accurately and efficiently controlling the deformation characteristics of a structure, the purpose of which is to achieve accurate and efficient topology optimization of a structure with spring deformation characteristics.

[0007] To achieve the above objectives, according to one aspect of the present invention, a topology optimization method for accurately and efficiently controlling structural deformation characteristics is proposed, comprising the following steps:

[0008] Finite element mesh generation is performed on the structure to be optimized, which has spring deformation characteristics, and design variables corresponding to the mesh are defined.

[0009] Based on the required spring stiffness coefficient and maximum deformation, the desired total structural strain energy is obtained, and the constraints on the total structural strain energy are determined. With the goal of minimizing the total structural volume, the first topology optimization model is established.

[0010] Topology optimization is performed based on the first topology optimization model to obtain the structural configuration and its total volume V. min ;

[0011] Based on the total volume V min Set the upper limit V′ of the overall structural volume constraint.min Thus, the total volume constraint of the structure is constructed. At the same time, the total strain energy constraint of the structure in the first topology optimization model is adopted. With the goal of minimizing the p-norm of the local deformation error under external force, the second topology optimization model is established.

[0012] Topology optimization is performed based on the second topology optimization model to obtain the final structural configuration.

[0013] As a further preferred embodiment, the first topology optimization model is expressed as follows:

[0014] find:x i i = 1, 2, ..., N ele

[0015]

[0016] st:KU=F

[0017]

[0018] 0≤x i ≤1

[0019] In the formula, N ele To determine the number of finite element meshes within the design domain, x i , v i These represent the design variables, physical variables, and volume corresponding to the i-th grid; J v The objective function is the total volume of the structure; KU = F is the finite element equilibrium equation, where F is the external force vector, U is the nodal displacement vector, and K is the overall structural stiffness matrix; g s Let k be the total strain energy constraint function of the structure. d U is the spring constant. max This represents the maximum deformation.

[0020] As a further preferred embodiment, the second topology optimization model is expressed as follows:

[0021] find:x i i = 1, 2, ..., N ele

[0022]

[0023] st:KU=F

[0024]

[0025] 0≤x i ≤1

[0026] In the formula, Ω Lu is a set of grid node numbers for the local area affected by external forces. j For set Ω L The displacement in the direction of the external force at the j-th node, where p is an even number greater than 1; J error Let g be the objective function. v This is the overall structural volume constraint function.

[0027] As a further preferred embodiment, for the external force array F, the direction of the external force is set to be parallel to only one direction in the three-dimensional Cartesian coordinate system, and all nodal forces in this array are equal, satisfying the following:

[0028]

[0029] 1 T F = k d u max

[0030] In the formula, D Force The array represents the numbering of the degrees of freedom at the point where the external force acts, length(D) Force ) represents the array D Force The length of F(D) Force ) represents the array D Force The corresponding nodal forces.

[0031] As a further preferred option, the design variable is a pseudo-density, which is obtained by density filtering and projection to obtain a physical variable that truly reflects the presence or absence of material in the mesh.

[0032] As a further preferred option, density filtering is performed using a spherical filter or a cylindrical filter, and projection is performed using a Heaviside step function.

[0033] As a further preferred option, based on the total volume V min Set the upper limit V′ of the overall structural volume constraint. min The calculation formula is:

[0034]

[0035] In the formula, ΔV represents the volume quantization error. This is the floor operator.

[0036] As a further preferred option, topology optimization is performed based on the first topology optimization model / second topology optimization model, including the following steps:

[0037] The displacement field is obtained by performing finite element analysis on the structure. Then, the sensitivity of the objective and constraint functions to the design variables is calculated. The design variables are updated using a gradient-based optimization algorithm. The above process is repeated until the convergence condition is met to obtain the structural configuration.

[0038] As a further preferred embodiment, the structure to be optimized with spring deformation characteristics is a spring, leaf spring, spring sheet, or integrated elastic support device.

[0039] According to another aspect of the present invention, a topology optimization system for precisely and efficiently controlling structural deformation characteristics is provided, comprising a processor for executing the aforementioned topology optimization method for precisely and efficiently controlling structural deformation characteristics.

[0040] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages:

[0041] 1. This invention cleverly utilizes mechanical principles to transform the maximum deformation constraint problem into a total structural strain energy constraint problem. Therefore, in topology optimization sensitivity analysis, only the sensitivity of each element's strain energy relative to its corresponding design variable needs to be calculated, unlike the global maximum displacement constraint problem which requires calculating the sensitivity of all node displacements relative to all design variables, significantly reducing computational load. Furthermore, it eliminates the need for repeated inversion of the overall stiffness matrix, resulting in a marked reduction in overall time consumption. Based on this, using statistical error description methods, the p-norm of the local deformation error under external force is used to characterize the maximum value of the deformation error. Minimizing this p-norm, with the total structural strain energy and volume as constraints, allows for further topology optimization. This approach achieves a topology with smaller maximum deformation error and more uniform deformation distribution while ensuring stable iteration and moderate computational load.

[0042] 2. This invention provides a topology optimization model that minimizes volume while constraining the total strain energy of the structure. This model can ensure the stability of the iterative process and quickly obtain a lightweight configuration that basically satisfies the spring deformation characteristics. Compared with the topology optimization method based on the global displacement p-norm, the computational load is significantly reduced, and the operation is more efficient.

[0043] 3. This invention provides a topology optimization model that takes the total strain energy and volume of the structure as constraints and minimizes the p-norm of the local deformation error under the action of external forces as the objective. Only a small number of nodal displacements are required to participate in the sensitivity calculation. Under the premise of low overall computational load, a topology configuration with reasonable material distribution, uniform deformation distribution and higher deformation accuracy can be obtained.

[0044] 4. The present invention mainly introduces physical constraints, which can be easily combined with various topology optimization methods such as variable density and level set. Although it adopts design variables based on grid cells, it can also be easily extended to design variables based on grid nodes. It is not limited by the type of topology optimization method and has good versatility.

[0045] 5. All objective and constraint functions included in this invention are explicit and differentiable, making it easy to calculate the objective and constraint functions and their design sensitivities. This facilitates the use of various gradient-based optimization algorithms for solving the problem and promotes widespread software integration.

[0046] 6. This invention uses spherical or cylindrical filters, which can obtain three-dimensional configurations with more flexible material distribution, as well as three-dimensional configurations with uniform cross-sectional characteristics. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of the topology optimization method for accurately and efficiently controlling structural deformation characteristics according to an embodiment of the present invention;

[0048] Figure 2 This is a schematic diagram of the design domain model of a leaf spring structure with load bias according to an embodiment of the present invention;

[0049] Figure 3 This is a three-dimensional schematic diagram of the topology optimization results of "strain energy constraint-volume minimization" in an embodiment of the present invention;

[0050] Figure 4 This is a three-dimensional schematic diagram of the topology optimization results for "strain energy and volume constraint - deformation error minimization" in an embodiment of the present invention;

[0051] Figure 5 This is a schematic diagram of the z-axis deformation of the topology optimization result of "strain energy constraint-volume minimization" in an embodiment of the present invention;

[0052] Figure 6 This is a schematic diagram of the z-axis deformation of the topology optimization results for "strain energy and volume constraint - minimization of deformation error" in an embodiment of the present invention;

[0053] Figure 7 This is a schematic diagram of the deformation of the mesh element in the z-direction within the deformation error minimization area according to an embodiment of the present invention;

[0054] Figure 8 This is a schematic diagram of the spring stiffness coefficient calculated under different external forces based on the topology optimization results of an embodiment of the present invention. Detailed Implementation

[0055] 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. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0056] This invention provides a precise and efficient topology optimization method for controlling structural deformation characteristics. First, the structure to be optimized, exhibiting spring deformation characteristics, is meshed using the finite element method (FEM). Design variables corresponding to the mesh are defined, and density filtering and step projection are used to obtain physical variables that accurately reflect the presence or absence of material in the mesh. Then, based on the spring stiffness coefficient and maximum deformation required for structural optimization design, the values ​​of the external force and maximum deformation in the finite element analysis are determined. Furthermore, according to the principle of virtual work in the generalized Hooke's theorem in solid mechanics (i.e., the total strain energy of the structure equals the virtual work done by the external forces), a topology optimization model is constructed with the total strain energy of the structure as a constraint and the minimization of the total structural volume as the objective. Then… Next, finite element analysis was used to obtain the displacement field, and the sensitivity of the objective and constraint functions and their relative design variables was calculated. Gradient optimization algorithms were used to update the design variables, and after convergence, the minimum structural volume that basically meets the requirements for deformation characteristics was obtained. Then, according to the volume value slightly larger than the above topology optimization result, with the total structural volume as the constraint, the total structural strain energy constraint was used, and the p-norm of minimizing the local deformation error under external force was used as the objective, a new topology optimization model was established. Finally, the sensitivity of all objective and constraint functions and their relative design variables was calculated again. Combined with finite element analysis, gradient optimization algorithms were used to continuously update the design variables. After convergence, the final topology configuration was obtained.

[0057] like Figure 1 As shown, the specific steps include the following:

[0058] Step 1: Perform finite element mesh generation on the structure to be optimized with specific deformation characteristics, define the design variables corresponding to the mesh, and obtain the physical variables that accurately reflect the presence or absence of material through density filtering and step projection.

[0059] Specifically, the structure to be optimized with specific deformation characteristics is partitioned, and N is obtained in the design domain. ele A finite element mesh is defined, and a pseudo-density x is defined corresponding to each mesh. i , (0≤xi≤1,i=1,2,...,N ele The design variable is denoted as ), and then the filter variable is calculated using formula (1).

[0060] ω j =max(1-||l j -l i || / R,0) (1)

[0061] In the formula, R is the filtration radius. v is the set of grid numbers within the filtering range. j With x j Let ω represent the volume of the j-th grid within the filter radius and the corresponding design variable, respectively.j This represents the weighting coefficient. When a spherical filter is used, the topology optimization result can have a relatively free material distribution in three-dimensional space. In this case, l i and l j Let ||l| represent the center coordinates of the i-th and j-th grids, respectively. j -l i || indicates l i and l j The distance between them; when a cylindrical filter is used, the topology optimization result can have a uniform cross-sectional characteristic, at which point l i and l j These represent the projected coordinates of the i-th and j-th grid centers within the cross section, respectively. Their coordinate values ​​along the stretching or sweeping path of the cross section do not participate in the distance ||l j -l i The calculation of ||.

[0062] Then, using a Heaviside step function similar to formula (2), the filtering variable is... Physical variables are obtained through projection calculations.

[0063]

[0064] In the formula, η and β represent the cutoff threshold and steepness of the Heaviside step projection function, respectively. Through the calculation of this formula, the physical variables gradually approach 0 or 1 in value, thus more realistically reflecting the presence or absence of material in the mesh, which facilitates more accurate subsequent finite element analysis.

[0065] Step 2: Based on the required spring stiffness coefficient and maximum deformation, define the total strain energy of the structure as the constraint function, and establish the first topology optimization model with the goal of minimizing the total volume of the structure.

[0066] Specifically, based on the design intent, the required spring stiffness coefficient k is obtained. d and maximum deformation u max Therefore, the magnitude of the resultant force of the external forces in the finite analysis can be calculated as k. d u max The desired total strain energy of the structure is Furthermore, based on the principle of virtual work in solid mechanics, the total strain energy constraint function of the structure is determined as shown in formula (3):

[0067]

[0068] In the formula, g sLet U be the total strain energy constraint function, U be the nodal displacement matrix obtained from finite element analysis, and K be the overall stiffness matrix of the structure. Then, using the total strain energy as a constraint and minimizing the total volume of the structure as the objective, a first topology optimization model is established, as shown in equation (4):

[0069]

[0070] In the formula, J v The objective function is the total volume of the structure; KU = F is the finite element equilibrium equation; F is the external force array, which is usually set to have the external force direction parallel to only one direction in the three-dimensional Cartesian coordinate system x, y, z, and all nodal forces in the array are equal, satisfying formula (5), and the resultant force satisfies formula (6):

[0071]

[0072] 1 T F = k d u max (6)

[0073] In the formula, D Force The array represents the numbering of the degrees of freedom at the point where the external force acts, length(D) Force ) represents the length of the array, u max This represents the maximum deformation in a certain direction (usually the same as the direction of the external force). By setting the above formula, the external force can be applied evenly to the structure.

[0074] By adopting the first topology optimization model, especially by setting the total strain energy constraint of the structure, we can ensure the stability of the topology optimization iteration process and ensure that the configuration obtained in each iteration has the same stiffness (or flexibility), and finally obtain the topology configuration that basically satisfies the desired spring deformation characteristics and has the smallest volume.

[0075] Step 3: Based on the first topology optimization model, calculate the total strain energy constraint function and its design sensitivity of the structure through the displacement field obtained by finite element analysis, update the design variables using a single-constraint gradient optimization algorithm, and obtain the current total volume of the structure after convergence.

[0076] Specifically, the displacement field of the structure is obtained by solving the finite element equilibrium equations, as shown in formula (7):

[0077] U=K -1 F (7)

[0078] In the formula, the overall structural stiffness matrix can be obtained by assembling the element stiffness matrices, as shown in formula (8):

[0079]

[0080] In the formula, pl is the material penalty factor, which is usually an integer not less than 1; E i Let α be the elastic modulus of the material in the i-th grid; α is a very small positive number to prevent physical variables from changing. A value of 0 results in a numerical singularity when inverting the overall stiffness matrix; L i The positioning matrix is ​​used to map the element stiffness matrix to the global stiffness matrix; k i This is the unit stiffness matrix (with an elastic modulus of 1) calculated when the i-th mesh is completely filled with material.

[0081] Furthermore, according to the chain rule, the sensitivity of the structural total volume objective function and the total strain energy constraint function to the design variables can be calculated using formulas (9) and (10):

[0082]

[0083] In the formula, the sensitivity of the total strain energy constraint function of the structure to the physical variables can be derived using the finite element equilibrium equations through the method of adjoint variables, as shown in formula (10):

[0084]

[0085] As can be seen from equation (11), the strain energy of each grid element only needs to be analyzed for sensitivity to its corresponding design variables, thus the overall computational load is relatively small.

[0086] In general, the sensitivity of physical variables relative to filter variables and the sensitivity of filter variables relative to design variables can be calculated using formulas (12) and (13), respectively:

[0087]

[0088] Then, the calculated objective and constraint functions and their sensitivity relative to the design variables are substituted into a gradient-based optimization algorithm based on the Moving Asymptotic Method (MMA) or the Optimization Criterion Method (OC) to update the design variables until convergence, resulting in a topological configuration that basically satisfies the spring deformation characteristics, while also having the minimum total structural volume V. min .

[0089] Step 4: Based on the above optimization results, set the overall volume constraint of the structure and continue to use the total strain energy constraint. Use the p-norm of the local deformation error of the external force application location to represent the maximum value of the deformation error. With the goal of minimizing this p-norm, establish the second topology optimization model.

[0090] Specifically, the upper limit V′ of the overall structural volume constraint is set with reference to the above optimization results. min Its value is slightly larger than the minimum total structural volume V obtained in step (3). minIt can usually be determined according to formula (14):

[0091]

[0092] In the formula, ΔV represents the volume quantization error, which can be taken as 0.05 times the initial total volume of the structure. This is the floor function operator. By setting a larger volume constraint, the direct use of V can be avoided. min Topology optimization makes it difficult to obtain new configurations. At the same time, a higher volume fraction is also conducive to obtaining more reasonable and robust topology optimization results, which can easily meet the minimum size constraints required by the manufacturing process.

[0093] The maximum value of the deformation error is represented by the p-norm of the local deformation error under external force, as shown in formula (15):

[0094]

[0095] In the formula, J error Let Ω represent the objective function. L u represents the set of mesh node numbers representing the local area affected by external forces. j This represents the displacement of the j-th node in a certain direction (usually consistent with the direction of the external force) within the set. The value of p is usually an even number greater than 1.

[0096] Furthermore, by taking the p-norm of the local deformation error at the point of application of external force as the optimization objective, it can be ensured that the spring deformation error at the point of application of external force is minimized in the topology optimization result; at the same time, with the structural volume and total strain energy as constraints, a second topology optimization model is established, as shown in formula (16):

[0097]

[0098] Step 5: Based on the second topology optimization model, calculate the new objective and constraint functions and their design sensitivity through the displacement field obtained by finite element analysis. Then, update the design variables again using the multi-constraint gradient optimization algorithm. After convergence, output the optimal topology configuration.

[0099] Specifically, the displacement field of the structure is obtained again by solving the finite element equilibrium equations according to formulas (7) and (8), and then the values ​​of the target and constraint functions are calculated according to formula (16).

[0100] Furthermore, the sensitivity of the objective function to the physical variables can be derived as shown in formula (17):

[0101]

[0102] In the formula, the sensitivity of nodal displacement to physical variables can be derived using the finite element equilibrium equations through the method of adjoint variables, as shown in formula (18):

[0103]

[0104] In the formula, U i Let I be the nodal displacement array of the i-th grid; j The introduced unit vector has columns equal to the number of degrees of freedom at all nodes, with the j-th row being 1 and the rest being 0; [K -1 I j ] i Indicates from K -1 I j Take the array of degrees of freedom of each node in the i-th grid from the array. From equations (17) and (18), it can be seen that the displacement u of a local node in a certain direction under the action of external force is... j All require sensitive calculations of all design variables, and in order to calculate K... -1 I j In addition to solving the finite element equilibrium equations, the global stiffness matrix inversion operation needs to be performed separately (this large-scale matrix inversion operation is time-consuming). Compared with using the global displacement p-norm as the optimization objective to achieve topology optimization, here we only need to solve the sensitivity of a small number of local node displacements to the design variables, thus greatly reducing the overall computational load and making the overall topology optimization operation more efficient.

[0105] Next, according to the chain method, combining formulas (12) and (13), the sensitivity of the overall structural volume constraint to the design variables can be calculated by formula (19):

[0106]

[0107] The sensitivity of the total strain energy constraint function of the structure to the design variables is still calculated using formula (10-13).

[0108] Finally, all the calculated objective and constraint functions and their sensitivity relative to the design variables are substituted into multi-constraint gradient optimization algorithms such as the Moving Asymptotic Method (MMA) for iterative solution until convergence, resulting in the final topology with accurate spring deformation characteristics.

[0109] The following are specific examples:

[0110] like Figure 2 The diagram shows the design domain model of a leaf spring structure with a biased load. The length, width, and height are 200mm, 30mm, and 2mm respectively. The lower left corner is located at the origin, and the length, width, and height coincide with the x, y, and z directions, respectively. The model is symmetrical. The entire structure is divided into 200×30×2 8-node regular hexahedral mesh elements. The desired spring stiffness coefficient k is... d =25N / mm, maximum deformation u max=4mm. A rectangular region of 40mm × 15mm on one side of the middle of the spring is designated as the non-design domain of the structure. A distributed load of resultant force F = 100N is applied to the surface of this region. The local deformation error p-norm of the external force is minimized on the mesh nodes at the white lines on the upper surface and both sides of this region. The surfaces at the left and right corners of the spring are fixed with a fixed area of ​​6mm × 2mm. A cylindrical filter is used with a density filtering radius R = 3mm, a density penalty coefficient pl = 3, p = 10 in the objective function, and a volume quantization error ΔV = 600mm. 3 The material selected for the design domain is titanium alloy with an elastic modulus of 110 MPa and α = 1.0 × 10⁻⁶. -9 The Poisson's ratio is 0.34; the location where the external force is applied is usually where specific equipment is installed, and it can be considered as an ultra-hard rigid material that hardly deforms. Therefore, its elastic modulus is set to 10 times the design domain, i.e., 1100 MPa. The maximum number of iterations is set to 170, the cutoff threshold η of the step projection function is 0.5, and the steepness β increases from 1 with the number of iterations. The specific changes are shown in formula (20):

[0111] loop∈[1, 170], loop∈Z + (20)

[0112] Figure 3 This is a 3D schematic diagram of the topology optimization result of "strain energy constraint-volume minimization", with a total structural volume of 3324 mm². 3 Therefore, based on the calculation results of formula (14), with 3600mm 3 As the upper limit of the total structural volume constraint, the total structural strain energy constraint is also used, with the goal of minimizing deformation error, resulting in the following... Figure 4 The topology optimization results are shown. As you can see, Figure 3 The structure within exhibits obvious small local hinges, while Figure 4 The structure is significantly more robust, and the material skeleton exhibits a certain degree of bending effect. The material distribution is more reasonable, making it easier to manufacture using methods such as laser cutting, and preventing the spring from breaking during processing.

[0113] Figure 5 , Figure 6The figures show the z-direction deformation diagrams for two topology optimization results: "strain energy constraint - volume minimization" and "strain energy and volume constraint - deformation error minimization". It can be seen that the former results in a larger variation in deformation along the y-direction at the location of the external force, while the latter shows a smaller variation, and the latter's average value is closer to the expected deformation. Table 1 shows the statistical results of the deformation data of the mesh elements at the location of deformation error minimization. The "strain energy and volume constraint - deformation error minimization" topology optimization result shows smaller average deformation, average deformation error, relative range percentage, and deformation standard deviation at the sampling location, indicating that minimizing deformation error as the objective function plays a significant role in topology optimization, thus obtaining a topology configuration with better spring deformation characteristics.

[0114] Table 1 shows the statistical data of mesh elements at the point where deformation error is minimized, based on the topology optimization results.

[0115]

[0116]

[0117] Figure 7 The average deformation of the element along the y-direction in the region of external force application shows that, compared with the topology optimization result of "strain energy constraint - volume minimization", the deformation curve of the topology optimization result of "strain energy and volume constraint - deformation error minimization" has a lower slope and is closer to the expected ideal deformation. Figure 8 The diagram shows the spring stiffness coefficients calculated based on the maximum or minimum deformation in the y-direction of the area affected by the external force under different external forces. It can be seen that the spring stiffness coefficients of the topology optimization results of "strain energy and volume constraint - deformation error minimization" are closer to the expected ideal stiffness coefficients, which fully demonstrates the accuracy of the design method.

[0118] This invention provides a precise and efficient topology optimization method for controlling structural deformation characteristics, and is a general systematic structural optimization design method. This method uses spherical or cylindrical filters, which can obtain both three-dimensional configurations with more flexible material distribution and those with uniform cross-sectional characteristics. Furthermore, it primarily introduces physical constraints, allowing for easy integration with various topology optimization methods such as variable density and level set optimization, without being limited by the type of topology optimization method, thus exhibiting good versatility. More importantly, by applying total strain energy as a constraint and minimizing volume as the objective, it can quickly obtain a lightweight configuration that basically satisfies the spring deformation characteristics while ensuring the stability of the iterative process. Compared with topology optimization methods based on the global displacement p-norm, it is more efficient. Finally, by minimizing the local deformation error p-norm under external force as the optimization objective, and using appropriate total structural volume constraints while retaining the total structural strain energy constraints, only a small number of nodal displacements are needed for sensitivity calculations. This allows for obtaining a topology configuration with reasonable material distribution, uniform deformation distribution, and higher deformation accuracy with relatively low overall computational load. This method opens up a new direction for structural optimization design that controls the deformation characteristics of springs. In the future, mature performance constraints such as maximum stress and manufacturing constraints such as minimum size can be introduced to obtain novel structures with better performance and manufacturability. It has significant engineering application value and will accelerate the technological transformation of structural optimization design in fields such as precision instruments and buffer devices.

[0119] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A topology optimization method for accurately and efficiently controlling structural deformation characteristics, characterized in that, Includes the following steps: Finite element mesh generation is performed on the structure to be optimized, which has spring deformation characteristics, and design variables corresponding to the mesh are defined. Based on the required spring stiffness coefficient and maximum deformation, the desired total structural strain energy is obtained, and the constraints on the total structural strain energy are determined. With the goal of minimizing the total structural volume, the first topology optimization model is established. Topology optimization is performed based on the first topology optimization model to obtain the structural configuration and its total volume. ; Based on total volume Set an upper limit for the overall structural volume constraint. Thus, the total volume constraint of the structure is constructed. At the same time, the total strain energy constraint of the structure in the first topology optimization model is adopted. With the goal of minimizing the p-norm of the local deformation error under external force, the second topology optimization model is established. Topology optimization is performed based on the second topology optimization model to obtain the final structural configuration; The first topology optimization model is represented as: In the formula, To determine the number of finite element meshes within the design domain, , , These represent the design variables, physical variables, and volume corresponding to the i-th grid; The objective function is the total volume of the structure. The finite element equilibrium equations are... For external forces to form an array, For nodal displacement arrays, This is the overall stiffness matrix of the structure; Let be the total strain energy constraint function of the structure. The spring constant is... For maximum deformation; The second topology optimization model is expressed as: In the formula, This is a set of grid node numbers for the local area affected by external forces. For set Inner Displacement in the direction of external force at each node It is an even number greater than 1; Let be the objective function. This is the overall structural volume constraint function.

2. The topology optimization method for accurately and efficiently controlling structural deformation characteristics as described in claim 1, characterized in that, For external force array Let the direction of the external force be parallel to only one direction in the three-dimensional Cartesian coordinate system, and let all nodal forces in this array be equal, and satisfy: In the formula, An array representing the degree of freedom numbers at the point where an external force acts. This indicates the array Length, This indicates the array The corresponding nodal forces.

3. The topology optimization method for accurately and efficiently controlling structural deformation characteristics as described in claim 1, characterized in that, The design variable is a pseudo density. After density filtering and projection, the design variable is used to obtain a physical variable that truly reflects the presence or absence of material in the mesh.

4. The topology optimization method for precisely and efficiently controlling structural deformation characteristics as described in claim 3, characterized in that, Density filtering is performed using a spherical or cylindrical filter, and projection is performed using the Heaviside step function.

5. The topology optimization method for accurately and efficiently controlling structural deformation characteristics as described in claim 1, characterized in that, Based on total volume Set an upper limit for the overall structural volume constraint. The calculation formula is: In the formula, For volume quantization error, This is the floor operator.

6. The topology optimization method for accurately and efficiently controlling structural deformation characteristics as described in claim 1, characterized in that, Topology optimization based on the first topology optimization model / second topology optimization model includes the following steps: The displacement field is obtained by performing finite element analysis on the structure. Then, the sensitivity of the objective and constraint functions to the design variables is calculated. The design variables are updated using a gradient-based optimization algorithm. The above process is repeated until the convergence condition is met to obtain the structural configuration.

7. The topology optimization method for accurately and efficiently controlling structural deformation characteristics as described in any one of claims 1-6, characterized in that, The optimized structure with spring deformation characteristics is a spring, leaf spring, spring sheet, or integrated elastic support device.

8. A topology optimization system for precisely and efficiently controlling structural deformation characteristics, characterized in that, Includes a processor, the processor being configured to execute the topology optimization method for precisely and efficiently controlling structural deformation characteristics as described in any one of claims 1-7.

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