Multi-material topological optimization and size control method based on isolated characteristic function

Abstract

The invention discloses a multi-material topological optimization and size control method based on an isolated characteristic function. The method comprises the following steps: discretizing a design space and initializing parameters; constructing a plurality of disjoint isolated characteristic functions by using a single optimization quantity, wherein each characteristic function corresponds to a sub-interval to distinguish different material phases; projecting the filtered topological density field to a normalized physical density field, and distributing upper and lower boundary characteristic functions for each subspace; the influence of the characteristics of the tiny structure is inhibited through two types of adjustment parameters of parameter zooming and translation; a weight factor is calculated, material attributes are weighted and represented through a plurality of clustering functions, and punishment is carried out by adopting an SIMP method; solving the balance equation to obtain a target function; and updating the design variables until convergence, outputting an optimal topological density field, and substituting the optimal topological density field into a post-processing function to draw a material distribution contour surface diagram. According to the method, distribution and configuration topological optimization design of different materials can be realized only by depending on a single design variable and an isolated characteristic function.

Description

Technical Field

[0001] This invention belongs to the field of structural optimization technology, specifically relating to an isolated multi-material topology optimization and size control method based on isolated characteristic functions. Background Technology

[0002] Topology optimization is a method of optimizing the spatial distribution of materials to meet specific design objectives under given design space, load conditions, boundary conditions, and material property constraints. Researchers have proposed various algorithms for topology optimization, including the isotropic material penalty method (SIMP), level set method, movable deformable component method, and bidirectional evolutionary structural optimization method. These different topology optimization methods have been widely applied in various research fields and have broad application prospects in engineering, covering multiple fields such as shipbuilding engineering, aerospace engineering, vehicle engineering, and civil engineering.

[0003] In traditional single-material topology optimization, the entire structure is composed of only one material, which to some extent limits the flexibility of the design space and may affect the performance of the optimization results. In contrast, multi-material topology optimization can significantly improve structural performance by configuring reasonable combinations of different materials, and is superior to single-material design in terms of design freedom, design space, and overall performance.

[0004] Existing multi-material topology optimization methods are mainly divided into two categories: univariate methods and multivariate methods. Currently, multivariate material interpolation methods based on DMO still dominate. A significant drawback of this type of method is the interdependence between design variables and the number of materials. The high-dimensional design space and high computational cost caused by the large number of design variables greatly reduce the convergence speed of the optimization algorithm. These limitations have prompted researchers to gradually explore univariate interpolation methods with obvious dimensionality reduction characteristics. However, reducing the number of discrete design variables often leads to unreasonable physical solutions (such as...). Figure 1 The material closure phenomenon reduces the manufacturability of the optimized structure, thus limiting its application in practical engineering. Furthermore, existing topology optimization methods lack reasonable length scale or feature size control, easily generating small-scale features such as excessively fine connections, sharp-corner structures, and isolated elements during the optimization process, severely impacting the manufacturability and engineering practicality of the structure. Summary of the Invention

[0005] This invention provides a multi-material topology optimization and size control method based on isolated characteristic functions. This method ensures that the number of design variables is independent of the number of material phases, avoiding the linear growth of design variables as the material type increases. It overcomes the non-physical design issues present in traditional single-variable design methods, such as intermediate material regions between rigid and flexible materials. It also eliminates the dependence of traditional methods on complex multi-material interpolation models, reducing the complexity of the optimization problem and avoiding the problem of easily getting trapped in local optima during the solution process. Furthermore, an effective small-scale feature control strategy is proposed to prevent structural features that are detrimental to manufacturing, such as excessively fine connections, sharp corners, and isolated units, from appearing during the optimization process.

[0006] To achieve the above objectives, the present invention adopts the following specific technical solution:

[0007] A multi-material topology optimization and size control method based on isolated eigenfunctions, the control method comprising: Step 1: Discretize the design space and initialize the parameters; Step 2: Construct multiple disjoint characteristic functions using a single optimization variable. Each characteristic function corresponds to a sub-interval to distinguish different material phases. Step 3: Obtain the filtered topological density field using the Helmholtz partial differential equation. The filtered topological density field is projected onto the normalized physical density field. Assign upper and lower bound characteristic functions to each subspace; adopt lower bound function Calculate the objective function C , C For structural flexibility; adopt upper bound function Calculate the constraint function, where the constraint function is the material volume / mass; Step 4: Calculate the weighting factors The SIMP method is used to The upper bound function is used to penalize the topological density field so that it approaches the discrete value 0 or 1, and the material properties are characterized by weighted summation of multiple clustering functions.

[0008] Step 5: Obtain the stiffness matrix K through finite element analysis, solve the equilibrium equation KU=F to obtain the displacement matrix U, and then obtain the structural compliance representing the objective function. C And calculate the sensitivity of the objective function and constraint functions to the design variables; Step 6: Update the design variables until the convergence condition is met, output the optimal topological density field, and substitute it into the post-processing function to draw the material distribution contour map.

[0009] Furthermore, in step three, a regularized Heaviside projection function is used to project the filtered topological density field onto the normalized physical density field. .

[0010] Furthermore, step two specifically includes: At any spatial location The range of design field changes Dividing the problem into non-overlapping sub-intervals, and using a univariate encoding method, the original discrete combinatorial optimization problem is mapped to a non-overlapping feature function representation. Phase material, as shown in equation (1): (1) Where Θ represents the number of candidate materials and the smoothness factor. and interval length factor control function shape, The value range is [0,1]; each material is associated with a single characteristic function, and there is no overlap between the characteristic functions, so as to realize the material selection in topology optimization. These characteristic functions can be regarded as optimization variables in existing methods. and The expression is defined as follows: (2) in, m The value is usually between 2 and 2.5.

[0011] Furthermore, step three includes: The formula for the Helmholtz PDE filtering method used is: (3) in, r Indicates the Helmholtz PDE filter radius; the unfiltered topological density field is... Indicated; the filtered topological density field is... express; Discretizing equation (3) and using uniform Neumann boundary conditions, we obtain the following system of linear equations: (4) in, This represents the unfiltered topological density field. This represents the filtered topological density vector field; Using the Heaviside projection function to Projected onto normalized density field : (5) The normalized Heaviside projection function is determined by the parameters β and γ Controls are used to manage the length dimensions of each material, thereby generating two types of boundary functions for the normalized physical topological density field: (6) (7) express The upper bound function, express The lower bound function.

[0012] 5. The multi-material topology optimization and size control method based on isolated characteristic functions as described in claim 4 is characterized in that the finite element method is used to discretize equation (3).

[0013] Furthermore, step four includes: Material properties are defined as a weighted sum of different characteristic functions: (8) (9) in, Indicates the first Weighting factors for each candidate material The value is in the range [0,1]. The weighting function for each material is defined as follows: (10) Among them, parameters It is obtained through formula (7); Substituting formula (10) into formulas (8) and (9), we get: (11) Using the SIMP method Punishment, so that It tends towards discrete values ​​of 0 or 1; The functions in equations (8), (9) and (11) g The definition is as follows: (12) Furthermore, step five includes: With structural flexibility as the optimization objective and the volume fraction of candidate materials as the constraint, the specific details are as follows: (13) in, The structural flexibility is represented by u, which is the objective function of the optimization problem; the displacement field of the structure is represented by u. Indicates design variables; Indicates the first θ The volume fraction of a material phase is used as a constraint function in the optimization problem. It is the first θ The preset maximum volume fraction of a material phase; |Ω| represents the volume of the design domain; g is at the boundary. The preset displacement vector; This represents the bilinear energy of the system. The linear load function of the system is defined as follows: (14) in, In space The virtual displacement field within; f represents the external force acting on the structure, and h represents the virtual displacement field at the boundary. Distributed traction force on the surface; Based on the chain rule, the objective function C Regarding design variables The sensitivity is expressed as: (15) objective function C Regarding weighting coefficients The derivative is defined as follows: (16) In equation (16) Determined by the differential rule; (17) From equation (12) we get ; Equation (17) can be rewritten as: (18) Based on this about The partial derivatives are derived as follows: (19) As described in equation (7), about The partial derivatives are expressed as: (20) Considering the Helmholtz PDE filtering equation given in equation (4), the filtering topological density field about The sensitivity is defined as follows: (twenty one) Isolated univariate feature function Regarding design variables The partial derivatives are expressed as: (twenty two) By substituting equations (18), (19), (20), and (22) into equation (15), the sensitivity of the objective function to the design variables is obtained: (twenty three) According to the chain rule, the volume constraint function Regarding design variables The sensitivity is expressed as: (twenty four) From the volume constraint function given by equation (13), the volume constraint about The partial derivatives yield: (25) lower bound function about The partial derivatives are expressed as: (26) and The expressions are shown in equations (21) and (22); by substituting equations (25), (26) and (21) into equation (24), the sensitivity of the constraint function with respect to the design variables is obtained: (27) Similarly, the mass constraint function is obtained. Regarding design variables The sensitivity is: (28) Furthermore, in step six, the design variables are updated based on the gradient optimization algorithm MMA.

[0014] Compared with the prior art, the technical solution of the present invention has the following beneficial effects: The multi-material topology optimization and size control method of this invention divides the global univariate design space into non-overlapping subdomains, each subdomain corresponding to a single characteristic function. Each characteristic function operates independently within its own subdomain while remaining within the overall design variable limits. Characteristic functions with different material annotations are generated in parallel and used as topology density variables for material selection. Each characteristic function is independent of other characteristic functions, and all possible characteristic function interactions are nonexistent. Therefore, any predefined material is only affected by its corresponding characteristic function and is not affected by materials in other characteristic functions. This allows the use of a single type of topology design variable to describe the distribution of various materials within the design domain. This solves the problem in existing technologies where calculating multi-material, multi-scale topology optimization problems requires different design variables for each material, and the design variables increase linearly with the number of materials, leading to increased computational costs.

[0015] The method described above presents a concise and efficient multi-material topology optimization model. It can achieve the allocation and configuration topology optimization design of different materials using only a single design variable and isolated characteristic functions, without the need for complex interpolation models, redundant design variables, or constraints. Furthermore, it effectively suppresses the small-scale characteristics of the optimization results. Numerical examples verify that this method significantly outperforms existing mainstream methods in terms of computational efficiency, convergence stability, and engineering feasibility, providing a new approach for multi-material topology optimization research and engineering applications. Attached Figure Description

[0016] Figure 1 A schematic diagram of a structure designed for the non-physical nature of material closure in traditional univariate methods; Figure 2 This is a flowchart of the multi-material topology optimization and size control method based on isolated characteristic functions of the present invention; Figure 3 Design domain and load boundary conditions for the right half of the 2D MBB beam; Figure 4 for Figure 3 Curves showing the variation of compliance value and volume fraction in topology optimization of 2D MBB beams; Figure 5 This is a comparison diagram of the optimization results of the single-variable multi-material topology optimization method of the present invention and existing methods; Figure 6 The design domain and load boundary conditions for a 2D bridge structure; Figure 7 A comparison diagram showing the optimized structural results of bridge structures using existing methods and the method of this invention; Figure 8 The results of multi-material topology optimization design for 2D bridge structures under cost constraints; Figure 9 Optimize the design of the GE bracket; Figure 10 The convergence curve during the optimization process; Figure 11a A 3D view of the multi-material topology optimization results for the engine mount; Figure 11b A top view of the multi-material topology optimization results for the engine mount; Figure 11c This is the main view of the multi-material topology optimization results for the engine mount; Figure 11d Rear view of the multi-material topology optimization results for the engine mount; Figure 11e-11g The distribution diagram of each phase material of the optimized engine mount; Figure 12 The design domain, loads, and boundary conditions for a 3D bridge structure; Figure 13 The iterative convergence plot of the objective function and volume fraction for a 3D bridge structure; Figure 14a A 3D model of the bridge structure; Figure 14b A top view of the 3D bridge structure; Figure 14c This is the front view of the 3D bridge structure; Figure 14d-14f This is a distribution diagram of the phase materials of the optimized 3D bridge. Detailed Implementation

[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0018] Existing technologies have the following drawbacks: 1. Existing multi-material topology optimization methods based on univariate interpolation models often result in non-physical intermediate materials between rigid and compliant materials. These intermediate materials cannot be completely removed in the final optimization design, excessively restricting the solution space, leading to suboptimal solutions, and limiting their practical applicability. 2. For traditional discrete composite interpolation models, such as the DMO method, each material requires at least one variable to characterize its presence or absence, meaning the number of design variables increases with the material type. Furthermore, in traditional topology optimization methods based on structured mesh finite element methods, the number of design variables is usually directly related to the number of elements or nodes in the finite element discretization. 3. In complex topology optimization problems such as multiphysics topology optimization, the complexity of the governing equations poses a significant challenge to sensitivity derivation. Although non-gradient algorithms (such as genetic algorithms) can avoid sensitivity derivation, the large number of design variables remains a problem to be overcome. 4. In the topology optimization process, structural configuration optimization usually involves a large number of degrees of freedom. Without proper control, small-scale structural features such as overly fine connections, sharp corners, and isolated units can easily be generated, making the optimized structure difficult to process and manufacture, thus severely limiting its practical application.

[0019] The problems this invention aims to solve are: 1. To achieve a non-overlapping distribution of multiple materials without increasing the dimensionality of design variables. This involves designing a single-variable characterization mechanism independent of material type, retaining the "low-dimensionality" advantage of single-variable methods while eliminating intermediate transitional materials between rigid and flexible materials through isolated characteristic functions, thus avoiding non-physical design. 2. To achieve length dimension control in multi-material topology optimization while ensuring clear material boundaries. This involves constructing a mapping mechanism of "isolated characteristic functions + PDE filtering + dynamic parameter adjustment," which suppresses small-scale features such as isolated units and slender connections without compromising the independence of material distribution, and supports differentiated scale constraints for different materials. 3. To simplify the computational logic of multi-material topology optimization, improve the optimization efficiency of large-scale engineering structures, avoid the dependence of traditional multi-material topology optimization on complex interpolation schemes (such as DMO and RMMI), directly map material properties through characteristic functions, reduce the computational complexity of sensitivity analysis and iterative solutions, and shorten convergence time.

[0020] This embodiment provides a multi-material topology optimization and size control method based on isolated characteristic functions, such as... Figure 2 As shown, the control method includes the following steps: Step 1: Discretize the design space and initialize the parameters; Step 2: Construct multiple disjoint characteristic functions using a single optimization variable. Each characteristic function corresponds to a sub-interval to distinguish different material phases. Step 3: Obtain the filtered topological density field using the Helmholtz partial differential equation. The filtered topological density field is projected onto the normalized physical density field using a regularized Heaviside projection function. Assign upper and lower bound characteristic functions to each subspace, and adopt lower bound function Calculate the objective function C , C For structural flexibility, a lower bound function is used to evaluate stiffness or strength performance; upper bound function Calculate the constraint function, which is the material volume / mass. Use the upper bound function to guide the optimization of volume / mass, thereby avoiding microstructures that are too large or too small. Step 4: Calculate the weighting factors By using multiple clustering functions to weight and characterize material properties, complex interpolation schemes are avoided. The SIMP method is employed to... A penalty is applied to make the topological density field approach discrete values ​​of 0 or 1, i.e., to push... It is developing towards discrete values ​​of 0 or 1; Step 5: Obtain the stiffness matrix K through finite element analysis, solve the equilibrium equation KU=F to obtain the displacement matrix U, and obtain the objective function representing the structural flexibility. C And calculate the sensitivity of the objective function and constraint functions to the design variables; Step 6: Update the design variables based on the gradient optimization algorithm MMA until the convergence condition is met, output the optimal topological density field, and substitute it into the post-processing function to draw the material distribution contour map.

[0021] In the above control method, step two specifically includes: At any spatial location The range of design field changes Dividing the problem into non-overlapping sub-intervals, and using a univariate encoding method, the original discrete combinatorial optimization problem is mapped to a non-overlapping feature function representation. Phase material, as shown in equation (1): (1) Where Θ represents the number of candidate materials and the smoothness factor. and interval length factor control function shape, The value range is [0,1]; each material is associated with a single characteristic function, and there is no overlap between the characteristic functions, so as to realize the material selection in topology optimization. These characteristic functions can be regarded as optimization variables in existing methods. and The expression is defined as follows: (2) in, m The value is usually between 2 and 2.5.

[0022] In the above control method, step three includes: The formula for the Helmholtz PDE filtering method used is: (3) in, r Indicates the Helmholtz PDE filter radius; the unfiltered topological density field is... Indicated; the filtered topological density field is... express; Discretizing equation (3) and using uniform Neumann boundary conditions, we obtain the following system of linear equations: (4) in, This represents the unfiltered topological density field. This represents the filtered topological density vector field; Using the Heaviside projection function to Projected onto normalized density field : (5) The normalized Heaviside projection function is determined by the parameters β and γ Controls are used to manage the length dimensions of each material, thereby generating two types of boundary functions for the normalized physical topological density field: (6) (7) express The upper bound function, express The lower bound function.

[0023] Through design parameters β We obtain suitable upper and lower bound functions. Using the upper bound function... Replace the original function It is used to evaluate stiffness or strength performance and effectively handle characteristic functions with fuzzy and uncertain features. It is used to calculate volume constraints, avoid excessively large or small microstructures, and thus achieve size control.

[0024] In the above control method, step four includes: Material properties are defined as a weighted sum of different characteristic functions: (8) (9) in, Indicates the first Weighting factors for each candidate material The value is in the range [0,1]. The weighting function for each material is defined as follows: (10) Among them, parameters It is obtained through formula (7); Substituting formula (10) into formulas (8) and (9), we get: (11) Using the SIMP method Punishment, so that It tends towards discrete values ​​of 0 or 1; The functions in equations (8), (9) and (11) g The definition is as follows: (12) Due to the use of a spatial partitioning strategy, the proposed scheme does not require complex dmo interpolation, and the optimization process only requires one type of design variable.

[0025] In the above control method, step five includes: With structural flexibility as the optimization objective and the volume fraction of candidate materials as the constraint, the specific details are as follows: (13) in, The structural flexibility is represented by u, which is the objective function of the optimization problem; the displacement field of the structure is represented by u. Indicates design variables; Indicates the first θ The volume fraction of a material phase is used as a constraint function in the optimization problem. It is the first θ The preset maximum volume fraction of a material phase; |Ω| represents the volume of the design domain; g is at the boundary. The preset displacement vector; This represents the bilinear energy of the system. The linear load function of the system is defined as follows: (14) in, In space The virtual displacement field within; f represents the external force acting on the structure, and h represents the virtual displacement field at the boundary. Distributed traction force on the surface; Based on the chain rule, the objective function C Regarding design variables The sensitivity is expressed as: (15) objective function C Regarding weighting coefficients The derivative is defined as follows: (16) In equation (16) Determined by the differential rule; (17) From equation (12) we get ; Equation (17) can be rewritten as: (18) Based on this about The partial derivatives are derived as follows: (19) As described in equation (7), about The partial derivatives are expressed as: (20) Considering the Helmholtz PDE filtering equation given in equation (4), the filtering topological density field about The sensitivity is defined as follows: (twenty one) Isolated univariate feature function Regarding design variables The partial derivatives are expressed as: (twenty two) By substituting equations (18), (19), (20), and (22) into equation (15), the sensitivity of the objective function to the design variables is obtained: (twenty three) According to the chain rule, the volume constraint function Regarding design variables The sensitivity is expressed as: (twenty four) From the volume constraint function given by equation (13), the volume constraint about The partial derivatives yield: (25) lower bound function about The partial derivatives are expressed as: (26) and The expressions are shown in equations (21) and (22); by substituting equations (25), (26) and (21) into equation (24), the sensitivity of the constraint function with respect to the design variables is obtained: (27) Similarly, the mass constraint function is obtained. Regarding design variables The sensitivity is: (28) To verify the effectiveness and correctness of the method proposed in the above embodiments, the following specific calculation examples are used for verification.

[0026] Example 1 Taking a two-dimensional MBB beam as an example, its design domain, loads, and boundary conditions are as follows: Figure 3 As shown. A unit load (f = 1) is applied at the top left corner of the design domain. The design domain is discretized using a 240 × 80 quadrilateral element mesh. This example studies the compliance minimization problem under different minimum length scale control conditions. The volume fraction of each material phase is set to 1 / 6 of the total volume of the design domain, i.e.

[0027] Table 1 Physical properties of candidate materials

[0028] Figure 4 The convergence curves of the objective function value versus material volume fraction during the multi-material topology optimization of the MBB beam structure are presented. Figure 4In the diagram, the objective function is the objective function, the volume fraction is the volume fraction, the iteration is the number of iterations, and the compliance is the compliance. It can be seen that the objective function converges to a stable value of 70.674 after only about 125 iterations, significantly lower than the 1800 iterations required by existing methods (ASADPOURE A, RAHMAN MM, NEJAT SA, et al. Topology optimization with multi-phase length-scale control [J]. International Journal of Mechanical Sciences, 2025, 291-292: 110086-), indicating that the proposed method has a significant improvement in optimization efficiency. Regarding the material volume fraction, all three candidate materials converge after about 100 iterations, and their convergence results all satisfy the preset volume constraints.

[0029] Figure 5 The MBB beam was demonstrated at three different filtering radii ( r =[3.5,3.5,3.5]、 r =[5.5,5.5,5.5] and r The graph shows the topology optimization results for the filter radius [3.5, 5.5, 7.5]. Each column corresponds to a different filter radius, and each row represents a different optimization method. Figure 5 Compared with the traditional topology optimization results a1-a3, the proposed method ( Figure 5 The optimized results obtained in (b1-b3) exhibit a clearer and more manufacturable material layout. However, these results still contain a large number of micropores of varying sizes and slender connecting rod structures, the physical meaning and engineering rationale of which are difficult to explain. In contrast, the proposed method, after introducing dimensional control ( Figure 5 The optimization results obtained from c1–c3 are smoother and more regular, with continuous and clear boundaries, while effectively eliminating fragmented small-scale features. Furthermore, from Figure 5 It can also be seen that the optimization results obtained using the method presented in this paper perform better in terms of mechanical rationality. Since MAT3 has the highest stiffness, it is mainly distributed in the boundary support region and the load application region to provide sufficient local stiffness. MAT2 is mainly distributed in secondary regions to achieve a smooth stiffness gradient. MAT1 is allocated to low-stress regions. This material distribution forms a clear and efficient load transfer path, thereby improving the overall structural stiffness and further validating the effectiveness of the proposed method.

[0030] Table 2 shows the comparison of the compliance of the optimized MBB beam structure under different filter radii. All compliance values ​​are calculated based on the semi-design domain. When the filter radius... r When the value is [3.5, 3.5, 3.5], the compliance value of the traditional method given in the prior art (ASADPOURE A, RAHMAN MM, NEJAT SA, et al. Topology optimization with multi-phase length-scale control[J]. International Journal of Mechanical Sciences, 2025, 291-292: 110086-) is 78.4, while the proposed single-variable method can reduce the structural compliance to 67.8372, a reduction of 13.5%. It should be noted that the compliance value of the proposed single-variable interpolation method with added size control is 68.8251, which is slightly higher than the version without size control, but still much lower than the traditional method, indicating that the optimized configuration obtained by this method can still have excellent stiffness performance while ensuring manufacturability. When the filter radius is... r When the coefficients are [3.5, 5.5, 7.5], the compliance value of the traditional method is as high as 87.9, while the proposed single-variable method still achieves a lower compliance value of 69.1193, a reduction of over 21.3%. Furthermore, the proposed method incorporating size control achieves a compliance value of 70.674, significantly lower than the traditional method. In summary, the proposed single-variable multi-material topology optimization method not only effectively improves structural stiffness, but also achieves a better balance between manufacturability and performance after incorporating the proposed size control strategy.

[0031] Table 2. Compliance values ​​of the MBB beam optimized structure under different filter radii

[0032] Table 3 below gives Figure 5 The volume constraint of each material phase in the optimized MBB beam structure is shown. It can be seen that, under different filter radii, the proposed univariate interpolation method (…) Figure 5 (b1)-(b3)) exhibit excellent consistency in material distribution, with the volume fractions of all material phases remaining stable between 16.57% and 16.68%, showing minimal fluctuation and meeting the set constraints. After introducing dimensional control ( Figure 5 (c1)-(c3)), the volume fractions of each material phase obtained by the proposed method remain highly consistent, concentrated at around 16.66%. These results show that the proposed method not only accurately satisfies the volume constraint, but also achieves consistency and stability of material distribution under different filter radii.

[0033] Table 3 Volume fraction of each material phase in MBB beams

[0034] Calculation example 2 This example studies the multi-material topology optimization design of a 2D bridge structure. The design domain, load conditions, and boundary conditions of the structure are as follows: Figure 6 As shown. Utilizing symmetry, only the right half of the structure is modeled and analyzed. The 2D bridge structure is subjected to two vertically downward concentrated loads, one of which has a magnitude of 2. f One force acts at the middle of the beam; the other concentrated force is... f Acting on the bridge span The design domain was discretized using a 100×100 finite element mesh. The volume fraction of each material phase was set to 1 / 6 of the total volume of the design domain, i.e. .

[0035] Figure 7 A comparative analysis of the optimization results of existing methods and the proposed method is presented for different filtering radii. Figure 7 a1 and b1 show the filter radius. r p Topology optimization results at a value of 3.5. Among the results obtained using the proposed method (e.g.) Figure 7 In section b1), material 3 is mainly distributed in the outer edge region of the arch structure, i.e., the region with higher stress. Material 2, with medium stiffness, is allocated to secondary critical regions to smooth and transfer loads. Material 1 is concentrated in non-critical load-bearing areas of the structure, serving as a connecting support. In contrast, existing methods (such as...) Figure 7 The material distribution obtained in a1) is quite chaotic and does not conform to the stress characteristics of the structure.

[0036] When the filter radius increases to r p =5.5 (e.g.) Figure 7 In (a2, b2), the material distribution of the two methods and r p The structure remains similar at 3.5, but the structural flexibility increases. Figure 7 a3 and b3 in the figure give the filter radius r For the design results under the condition [3.5, 5.5, 7.5], although all volume constraints are satisfied, there are significant differences in the material distribution characteristics between the two methods. Existing research results (such as...) Figure 7 In (a3), the morphology of material 1 undergoes significant changes, with two relatively large material aggregation regions appearing. The proposed method yields optimized results (…). Figure 7 b3) avoids this problem, not only maintaining better material continuity, but also having a structural topology configuration that is almost identical to the results under the same filtering radius.

[0037] Table 4 below compares the compliance values ​​of the optimized 2D bridge structure under different filter radii. Numerical results show that, compared with traditional topology optimization methods, the compliance values ​​obtained by the proposed single-variable method are significantly lower than those obtained by traditional methods under different filter radii, achieving superior structural mechanical performance. It is worth noting that... r Under the condition of [3.5, 5.5, 7.5], the proposed method achieves a significant reduction in flexibility, which contrasts sharply with the traditional trend in topology optimization where "the stronger the design space constraint, the higher the flexibility usually becomes." Furthermore, the volume fraction results presented in Table 5 show that the volume fractions of each phase material in the optimized bridge structure all meet the preset volume constraints.

[0038] Table 4 Different minimum filter radii Performance of the optimized bridge structure

[0039] Table 5 Figure 7 Volume fraction of each material phase in a 2D bridge structure

[0040] Cost constraints were introduced during the optimization process, and the topology optimization design of 2D bridge structures was studied based on the proposed method. Figure 8 Three different combinations of filter radii are presented under cost constraints. r p =3.5、 r p =5.5 and r The results of multi-material topology optimization for 2D bridges with coordinates [3.5, 5.5, 7.5]. Comparative analysis shows that the optimized structure's topology configuration changes significantly compared to the optimization results without cost constraints, with material 2 largely replacing material 3 in the outer region of the arch structure.

[0041] Tables 6 and 7 present the structural flexibility and material volume distribution of the 2D bridge optimization design under cost constraints. The results show that the flexibility of all configurations increases after introducing cost constraints. As shown in Table 6, the flexibility values ​​obtained by the proposed method are significantly lower than those of the traditional topology optimization method under all different filter radius combinations. This indicates that the proposed univariate method can achieve better material distribution under cost constraints, improving the overall structural stiffness while maintaining economy, especially in mixed filter radius configurations. r Under [3.5, 5.5, 7.5] conditions, the proposed method can still maintain a relatively low flexibility.

[0042] Table 6. Different filter radii Figure 8 Performance of the optimized bridge structure

[0043] Table 7 Figure 8 Volume fraction of each part of the bridge structure

[0044] Furthermore, as shown in Table 7, the proposed method achieves a more reasonable material distribution while strictly satisfying all specified volume constraints. In the results of traditional topology optimization methods, although the volume percentage of each material phase is strictly controlled within preset values, the material layout fails to further improve structural stiffness. In contrast, under the same cost constraints, the proposed method exhibits a more reasonable material distribution and topological configuration, resulting in better structural stiffness performance. For example, in... Figure 8 In b1, the actual volume fractions of the three material phases are 10.13%, 14.21%, and 10.77%, respectively. These results not only meet the set cost constraints for each material phase, but also effectively reduce the structural flexibility by replacing the highest stiffness material with a medium-stiffness material in the outer region of the structure. Thus, while controlling costs, the structural stiffness is maximized by optimizing the material distribution. Therefore, it can be considered that the proposed method can strictly meet the set constraints and achieve a better balance between cost constraints and mechanical performance.

[0045] Calculation example 3 This section further verifies the effectiveness and universality of the proposed method for complex practical engineering problems. Using the isolated characteristic function interpolation method proposed in this invention, a GE support structure and a 3D bridge structure are selected as representative engineering cases for multi-material topology optimization research. The candidate materials are aluminum, iron, and titanium, with volume fraction constraints for each phase set to 0.2, 0.15, and 0.1, respectively. The material parameters are listed in Table 8.

[0046] Table 8 Material performance parameters used in three-dimensional bridge engineering structures

[0047] GE support structure design domain 2, external loads and boundary conditions, such as Figure 9 As shown. Design domain Ω D Discretization was performed using a structured mesh of 89×31×54. Based on actual working conditions, external forces were applied to the two mounting holes at the top of the support, while the four mounting holes 1 at the bottom were subject to fixed displacement constraints, forming a fixed displacement region.

[0048] Figure 10The convergence characteristics of the objective function and volume fraction of the GE support in the multi-material topology optimization process are demonstrated. The objective function gradually decreases with increasing iteration number, finally converging to 0.038878 at the 88th iteration. Simultaneously, the volume fractions of the three materials stabilize after approximately 60 iterations, satisfying the set volume constraints. These results indicate that the proposed optimization algorithm has strong convergence stability on this example.

[0049] The results of GE stent multi-material topology optimization are as follows: Figure 11a As shown in -g, the optimized structure exhibits smooth boundaries and clear material interfaces. The red titanium material is mainly concentrated in critical load-bearing or high-stress areas, utilizing its high elastic modulus to provide higher local stiffness. The green aluminum material is distributed in non-critical areas, providing necessary support and connection. The blue iron material is mainly located between the titanium and aluminum regions. The proposed method achieves a reasonable material distribution, effectively enhancing the load transfer path.

[0050] Next, taking the 3D bridge structure as the design object, we used unstructured meshes for finite element discretization. Figure 12 The design domain, loads, and boundary conditions are given. The bridge dimensions are 400×80×80mm. The entire bridge is divided into the design domain. Non-design domain Uniform force F = 16667 Pa along The negative direction of the axis acts on the non-design area of ​​the bridge. Load action region Г t Above. Fixed displacement region at both ends below the bridge. v The boundary conditions are defined as follows. The finite element mesh has 63,463 elements and 14,267 nodes. Based on the proposed multi-material topology optimization method, the flexibility of the bridge structure is minimized while satisfying volume constraints by rationally allocating candidate materials at different spatial locations of the bridge.

[0051] Figure 13The convergence curves of the objective function and the volume fractions of the three materials in the 3D bridge structure optimization process are presented. It can be seen that the objective function increases sharply at the beginning, but decreases sharply after approximately the 35th iteration, eventually stabilizing. Furthermore, the volume fraction of aluminum decreases rapidly in the early stages of iteration, then stabilizes. The other two materials show a trend of first decreasing and then increasing. Around the 50th iteration, the volume fractions of the three materials essentially converge.

[0052] Figure 14a The `-f` option specifies the material layout of the 3D bridge obtained through multi-material topology optimization. From different perspectives, it can be seen that the high-stiffness titanium material is concentrated in stress-concentrated areas such as the boundaries and load application points. Iron, with medium stiffness, is mainly distributed along the main load transmission paths. In contrast, the low-stiffness aluminum material occupies non-critical areas, primarily serving a supporting and connecting role. The optimized 3D bridge structure configuration and material distribution improve structural stiffness, and the interfaces between different phase materials are clear and smooth, meeting the requirements of additive manufacturing and facilitating actual processing and manufacturing.

[0053] Obviously, those skilled in the art can make various modifications and variations to the embodiments of the present invention without departing from the spirit and scope of the invention. Therefore, if these modifications and variations fall within the scope of the claims of the present invention and their equivalents, the present invention also intends to include these modifications and variations.

[0054] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., 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 multi-material topology optimization and size control method based on isolated characteristic functions, characterized in that, include: Step 1: Discretize the design space and initialize the parameters; Step 2: Construct multiple disjoint characteristic functions using a single optimization variable. Each characteristic function corresponds to a sub-interval to distinguish different material phases. Step 3: Obtain the filtered topological density field using the Helmholtz partial differential equation. The filtered topological density field is projected onto the normalized physical density field. Assign upper and lower bound characteristic functions to each subspace; adopt lower bound function Calculate the objective function C , C For structural flexibility; adopt upper bound function Calculate the constraint function, where the constraint function is the material volume / mass; Step 4: Calculate the weighting factors The SIMP method is used to A penalty is applied to make the topological density field approach discrete values ​​of 0 or 1, and the material properties are characterized by weighted summation using multiple clustering functions; Step 5: Obtain the stiffness matrix K through finite element analysis, solve the equilibrium equation KU=F to obtain the displacement matrix U, and then obtain the structural compliance representing the objective function. C And calculate the sensitivity of the objective function and constraint functions to the design variables; Step 6: Update the design variables until the convergence condition is met, output the optimal topological density field, and substitute it into the post-processing function to draw the material distribution contour map.

2. The multi-material topology optimization and size control method as described in claim 1, characterized in that, In step three, the filtered topological density field is projected onto the normalized physical density field using the regularized Heaviside projection function. .

3. The multi-material topology optimization and size control method as described in claim 2, characterized in that, Step two specifically includes: At any spatial location The range of design field changes Dividing the problem into non-overlapping sub-intervals, and using a univariate encoding method, the original discrete combinatorial optimization problem is mapped to a non-overlapping feature function representation. Phase material, as shown in equation (1): （1） Where Θ represents the number of candidate materials and the smoothness factor. and interval length factor control function shape, The value range is [0,1]; each material is associated with a single characteristic function, and there is no overlap between the characteristic functions, so as to realize the material selection in topology optimization. These characteristic functions can be regarded as optimization variables in existing methods. and The expression is defined as follows: （2） in, m The value is usually between 2 and 2.

5.

4. The multi-material topology optimization and size control method as described in claim 3, characterized in that, Step three includes: The formula for the Helmholtz PDE filtering method used is: （3） in, r Indicates the Helmholtz PDE filter radius; the unfiltered topological density field is... Indicated; the filtered topological density field is... express; Discretizing equation (3) and using uniform Neumann boundary conditions, we obtain the following system of linear equations: （4） in, This represents the unfiltered topological density field. This represents the filtered topological density vector field; Using the Heaviside projection function to Projected onto normalized density field : （5） The normalized Heaviside projection function is determined by the parameters β and γ Controls are used to manage the length dimensions of each material, thereby generating two types of boundary functions for the normalized physical topological density field: （6） （7） express The upper bound function, express The lower bound function.

5. The multi-material topology optimization and size control method as described in claim 4, characterized in that, The finite element method is used to discretize equation (3).

6. The multi-material topology optimization and size control method as described in claim 5, characterized in that, Step four includes: Material properties are defined as a weighted sum of different characteristic functions: （8） （9） in, Indicates the first Weighting factors for each candidate material The value is in the range [0,1]. The weighting function for each material is defined as follows: （10） Among them, parameters It is obtained through formula (7); Substituting formula (11) into formulas (9) and (10), we get: （11） Using the SIMP method Punishment, so that It tends towards discrete values ​​of 0 or 1; The functions in equations (8), (9) and (11) g The definition is as follows: （12）。 7. The multi-material topology optimization and size control method as described in claim 5, characterized in that, Step five includes: With structural flexibility as the optimization objective and the volume fraction of candidate materials as the constraint, the specific details are as follows: (13) in, The structural flexibility is represented by u, which is the objective function of the optimization problem; the displacement field of the structure is represented by u. Indicates design variables; Indicates the first θ The volume fraction of a material phase is used as a constraint function in the optimization problem. It is the first θ The preset maximum volume fraction of a material phase; |Ω| represents the volume of the design domain; g is at the boundary. The preset displacement vector; This represents the bilinear energy of the system. The linear load function of the system is defined as follows: (14) in, In space The virtual displacement field within; f represents the external force acting on the structure, and h represents the virtual displacement field at the boundary. Distributed traction force on the surface; Based on the chain rule, the objective function C Regarding design variables The sensitivity is expressed as: (15) objective function C Regarding weighting coefficients The derivative is defined as follows: (16) In equation (16) Determined by the differential rule; (17) From equation (12) we get ; Equation (17) can be rewritten as: (18) Based on this about The partial derivatives are derived as follows: (19) As described in equation (7), about The partial derivatives are expressed as: (20) Considering the Helmholtz PDE filtering equation given in equation (4), the filtering topological density field about The sensitivity is defined as follows: (21) Isolated univariate feature function Regarding design variables The partial derivatives are expressed as: （22） By substituting equations (18), (19), (20), and (22) into equation (15), the sensitivity of the objective function to the design variables is obtained: （23） According to the chain rule, the volume constraint function Regarding design variables The sensitivity is expressed as: （24） From the volume constraint function given by equation (13), the volume constraint about The partial derivatives yield: （25） lower bound function about The partial derivatives are expressed as: (26) and The expressions are shown in equations (21) and (22); by substituting equations (25), (26) and (21) into equation (24), the sensitivity of the constraint function with respect to the design variables is obtained: （27） Similarly, the mass constraint function is obtained. Regarding design variables The sensitivity is: （28）。 8. The multi-material topology optimization and size control method as described in any one of claims 1-7, characterized in that, In step six, the design variables are updated based on the gradient optimization algorithm MMA.

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