A design method for variable-section beams based on multi-scale finite elements

Abstract

The present invention relates to a variable cross-section beam design method based on multi-scale finite elements, the method comprising: firstly dividing the beam model into macro-scale grids, using this as the macro-unit of the beam model; then establishing a representative volume element (RVE) at the integration point of the macro-beam unit, and dividing the RVE into meso-scale grids, using this as the meso-unit of the beam structure; then establishing periodic boundary conditions and equilibrium equations between the macro- and meso-units; then establishing the objective function and volume constraint of topology optimization on the meso-units, and optimizing; finally obtaining the optimized beam structure. The present invention improves the traditional single-scale full finite element topology optimization method of non-homogeneous beams, which has the problems of a large number of units, long calculation time, many details, and a single optimization result. The method has the advantages of simple design, short calculation time, multivariate optimization results, and wide applicability.

Description

Technical Field

[0001] The invention relates to the field of variable cross-section beam design, and in particular to a variable cross-section beam design method based on multi-scale finite elements. Background Art

[0002] The multiscale finite element method is a numerical calculation method used to simulate and analyze the mechanical behavior of materials and structures with multiple scales. Its basic idea is to first divide the entire material or structural system into different scales, then apply the finite element method at each scale, and establish the connection between the scales through certain coupling methods such as localization and homogenization. Multiscale methods can be divided into two categories according to modeling and calculation strategies: hierarchical multiscale analysis methods and concurrent multiscale analysis methods. The multiscale finite element method has important applications in many fields. For example, in the engineering field, the multiscale finite element method can be used to design and optimize structures, improve the performance and reliability of structures, and improve the quality and life of products. In the field of materials, it can be used to study and predict the mechanical properties of complex materials such as composite materials and multiphase materials.

[0003] Direct FE 2 The Direct FE method is a concurrent multiscale finite element method. 2 From the energy perspective, the model applies the macroscopic strain field to the mesoscopic representative volume element (RVE) through periodic boundary conditions to achieve the scale transition from macro to meso. Then, based on the Hill-Mandel principle, the RVE is stiffened so that the total energy of the RVE at the Gaussian point in the macro unit is equal to the total energy of the macro unit, thereby achieving the scale transition from meso to macro. Finally, the finite element solution is used to obtain the solutions of the macro and meso scales at the same time. This concurrent multi-scale analysis method only requires one finite element solution, without the need for traditional FE. 2 The nested iterative solution of the method improves the computational efficiency.

[0004] A variable-section beam refers to a beam structure in which the cross-sectional shape and size of the beam gradually change along the axial direction of the beam. Variable-section beams are widely used in bridge design, construction engineering, aerospace and other fields. By changing the shape and size of the beam cross section, a better distribution of materials can be achieved according to the stress conditions of the beam, thereby maximizing the utilization efficiency of the materials. In terms of engineering applications, variable-section beams can reduce the weight of the structure while meeting requirements such as structural stiffness, while also saving materials.

[0005] Structural topology optimization is an effective structural layout method. By optimizing the structural layout, lightweight design of the structure can be achieved. At present, topology optimization has been successfully applied in aerospace, automobile, construction and medical fields. In recent years, structural topology optimization has been considered one of the most challenging and most cost-effective tasks in structural design.

[0006] In the optimization design of non-homogeneous beams, the traditional optimization method is mainly to design through a single-scale topological optimization method. First, a non-homogeneous beam is modeled. Taking a cantilever beam as an example (such as Figure 1 As shown in the figure), the left end of the beam is fixed, and a downward force F is applied to the right end of the beam. Then, the beam model is volume constrained (the mass or volume retained during optimization is expressed as V f ), set up the objective function (such as minimizing strain energy, expressed as min C(ρ)), then perform topology optimization (variable density method) step by step iteration, and finally obtain the optimized beam model under volume constraint control. The optimized beam presents a truss structure (such as Figure 2 As shown in ). This single-scale topology optimization method requires that the beam model be fully discretized. In order to ensure the accuracy of the finite element calculation, the more discrete units there are, the higher the accuracy will be, but the corresponding calculation time and memory usage will also be higher, which obviously does not meet the original intention of rapid design. Variable-section beams are easier to implement the "equal strength" principle, which can save materials to the maximum extent while ensuring structural strength, improve the stiffness and bearing capacity of the beam, and improve the performance of the overall structure by designing a reasonable variable-section shape (such as Figure 3 shown).

[0007] Based on the above statements, in order to improve the topological optimization design of heterogeneous beam structures, the present invention proposes a multi-scale finite element method (Direct FE 2 The method of topological optimization of local RVE can greatly reduce the number of finite element units, save calculation time, and realize the design of variable cross-section beams, which is far superior to the truss beam structure obtained by traditional topological optimization. Summary of the invention

[0008] In view of the above-mentioned problems, the present invention proposes a variable-section beam design method based on multi-scale finite elements.

[0009] To achieve the above object, the present invention provides the following technical solutions:

[0010] Terminology explanation:

[0011] Multi-scale finite element: refers to a numerical calculation method that establishes finite element models at different scales of structures or materials to simulate the mechanical behavior of materials and structures.

[0012] Direct FE 2 Method: A concurrent multiscale finite element method. Direct FE 2 From the energy perspective, the model applies the macroscopic strain field to the mesoscopic representative volume element (RVE) through periodic boundary conditions. Based on the Hill-Mandel principle, the RVE is volume-scaled so that the total energy of the RVE at the Gaussian point in the macroscopic unit is equal to the total energy of the macroscopic unit, and the scale transition from the mesoscopic structure to the macroscopic structure is established. Finally, the macro and mesoscopic structures are solved simultaneously through finite element solution. This concurrent multi-scale analysis method only requires one finite element solution, without the need for traditional FE 2 The nested iterative solution of the method improves the computational efficiency.

[0013] Topology optimization: It is a method to calculate the optimal distribution of materials / structures under given constraints. By optimizing the layout of materials / structures, lightweight design of structures can be achieved.

[0014] A variable cross-section beam design method based on multi-scale finite elements, the method comprising the following steps:

[0015] S01, dividing the beam model into macro-scale grids, which are used as macro-units of the beam model;

[0016] S02, establishing an RVE model at the integration point of the macro unit, and dividing the RVE into mesoscopic scale grids, which are used as the mesoscopic units of the beam structure;

[0017] S03, then establish the periodic boundary conditions and equilibrium equations between the macroscopic and mesoscopic units;

[0018] S04. Establishing the objective function and volume constraint of topology optimization on the mesoscopic unit so as to optimize the mesoscopic structure of the beam structure;

[0019] S05. Finally, through multiple optimization iterations, the optimized beam structure was obtained, which met the design requirements and constraints and had good mechanical properties and economy.

[0020] As a further technical solution of the present invention, in step S01, the step of dividing the beam model into macro-scale grids to form macro-units of the beam model includes:

[0021] First, the beam model is modeled, and the beam unit is used to mesh the beam model to obtain a macro-scale grid, which is used as the macro-unit of the beam model. During the meshing process, it is necessary to select an appropriate mesh size and unit type to ensure the accuracy and reliability of the calculation results. Generally, the smaller the size of the beam unit, the more accurate the calculation result, but it will also increase the complexity and time cost of the calculation. After the beam unit is meshed, the corresponding boundary conditions need to be applied, including the fixed end, free end and load of the beam, which need to be set according to the actual situation. The beam unit is given a preset Young's modulus, which is a Young's modulus that can be ignored compared to the actual material, for example: 1×10 -9 E 0 , where E 0 is the actual Young's modulus of the material.

[0022] As a further technical solution of the present invention, in step S02, the steps of establishing an RVE model at the integration point of the macro unit and dividing the RVE into mesoscopic scale grids to use this as the mesoscopic unit of the beam structure include:

[0023] An RVE model is established at the integration point of the macro unit to capture the microscopic characteristics of the beam structure; a plane model is established for the two-dimensional RVE, and a corresponding solid model is established for the three-dimensional RVE. The corresponding two-dimensional or three-dimensional units are used to perform finite element meshing on the RVE to obtain a mesoscopic scale mesh, and the actual material parameters of the RVE, including Young's modulus, Poisson's ratio, etc., are assigned to the mesoscopic unit of the beam structure. During the meshing process, it is necessary to select an appropriate mesh size and unit type to ensure the accuracy and reliability of the calculation results.

[0024] As a further technical solution of the present invention, in step S03, the periodic boundary condition is as shown in Formula 1:

[0025] Formula 1:

[0026]

[0027] in, and Respectively represent the displacement vector of the right node of RVE and the corresponding displacement vector of the left node, and They represent the displacement vector of the upper node of RVE and the corresponding displacement vector of the lower node, L x and L y denote the length of RVE along the x and y directions, d i represents the node displacement vector of the macro unit, N i,x and N i,t They represent the derivatives of the macro unit shape function with respect to the x-coordinate and the y-coordinate respectively;

[0028] In multi-scale analysis, since the constitutive relation of the macroscopic unit cannot be directly obtained, the internal virtual work of the macroscopic unit can be converted into the internal virtual work of the mesoscopic unit according to the Hill-Mandel principle, thus establishing the Direct FE of the beam. 2 Multiscale finite element model;

[0029] Formula 2:

[0030] Among them, δW int is the internal virtual work of RVE, w α and J α are the weight and Jacobian at the Gaussian integration point α, A is the cross-sectional area of ​​the beam, is the virtual strain of RVE, is the stress of RVE.

[0031] As a further technical solution of the present invention, in step S03, the stiffness of the RVE is scaled based on the Hill-Mandel principle during the establishment of the equilibrium equation, and the scaling coefficient is determined according to the following formula:

[0032]

[0033] Where w is the scaling factor, w α is the weight of the Gaussian integration point α where RVE is located, J α is the value of the Jacobian at the Gaussian integration point α where RVE is located, A is the cross-sectional area of ​​the beam, V α is the volume of RVE at Gaussian integration point α. The RVE stiffness can be scaled by scaling the volume of RVE. For two-dimensional problems, the RVE stiffness can be scaled by enlarging the thickness of RVE, with a magnification factor of w. For three-dimensional problems, the RVE stiffness can be scaled by enlarging RVE in three directions in equal proportions, with a magnification factor of

[0034] As a further technical solution of the present invention, in step S04, the objective function and volume constraint of topology optimization are established on the mesoscopic unit before optimization. When maximizing the structural stiffness, the objective function is the total strain energy of the structure, and the volume constraint is the percentage of the volume retained after structural optimization to the total volume of the structure before optimization. The purpose of the volume constraint is to limit the material usage of the beam structure to meet the cost and manufacturing constraints in actual engineering. The volume constraint is given by the designer before optimization based on actual problems. After establishing the objective function and volume constraint, a topology optimization algorithm, such as SIMP (Solid Isotropic Material with Penalization) or BESO (Bi-directional Evolutionary Structural Optimization), is used to optimize the beam structure; the topology optimization algorithm can automatically search for the optimal topological structure of the beam structure to meet the requirements of the objective function and volume constraint. Through optimization, the optimal mesoscopic structure of the beam structure can be obtained to improve the stiffness, strength and other performance indicators of the beam structure.

[0035] Compared with the prior art, the present invention has the following beneficial effects:

[0036] The topology optimization method proposed by the present invention through concurrent multi-scale finite element is based on the mesoscopic scale of the heterogeneous beam, and a mesoscopic finite element model is established. The scale transformation between macro and mesoscopic scales is realized through periodic boundary conditions and stiffness scaling based on the Hill-Mandel principle, and the local structure of the beam is optimized according to the objective function and volume constraints. The present invention improves the problems of large number of units, long calculation time, and single optimization results brought about by the traditional single-scale full finite element topology optimization method for heterogeneous beams. The method has the advantages of simple design, short calculation time, multivariate optimization results, and wide applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 Schematic diagram of a cantilever beam with concentrated force at one end.

[0038] Figure 2 This is the topology optimization result diagram of a traditional single-scale cantilever beam.

[0039] Figure 3 This is the topology optimization result diagram of the variable cross-section cantilever beam based on multi-scale finite element.

[0040] Figure 4 A flow chart for designing a variable cross-section beam based on multi-scale finite elements is provided in the present invention.

[0041] Figure 5 The two-dimensional Direct FE provided by the present invention 2 Diagram of the cantilever beam model.

[0042] Figure 6 The two-dimensional Direct FE provided by the present invention 2 Result diagram of cantilever beam variable cross-section topology optimization.

[0043] Figure 7 This is a diagram of the traditional full finite element topology optimization results of the two-dimensional cantilever beam provided by the present invention.

[0044] Figure 8 The three-dimensional Direct FE provided by the present invention 2 Diagram of the simply supported beam model.

[0045] Fig. 9 The three-dimensional Direct FE provided by the present invention 2 Result diagram of topology optimization of simply supported beam with variable cross-section.

[0046] Fig.10 This is a diagram of the traditional full finite element topology optimization results of the three-dimensional simply supported beam provided by the present invention. DETAILED DESCRIPTION

[0047] The technical solution of the present invention is further described in detail below in conjunction with specific implementation methods.

[0048] See also Figure 4 The embodiment of the present invention provides a variable cross-section beam design method based on multi-scale finite elements, the method comprising the following steps:

[0049] S01, dividing the beam model into macro-scale grids, which are used as macro-units of the beam model;

[0050] Specifically, the beam model is first modeled, and the beam model is meshed using beam elements to obtain a macro-scale grid, which is used as the macro-unit of the beam model. Here, for the sake of illustration, the beam structure is only divided into four beam elements. The left end of the model is fixed, and a downward concentrated force is applied to the right end of the model. The beam element is given a Young's modulus that is negligible compared to the actual material. Here, the Young's modulus is set to 1×10 -9 MPa;

[0051] S02. Establish the RVE model at the integration point of the macro unit, and use the plane stress unit to mesh the RVE, which is the mesoscopic unit of the beam structure. And give the unit actual material parameters, among which the Young's modulus is 7×10 4 MPa, Poisson’s ratio is 0.3;

[0052] In this embodiment, an RVE model is established at the integration point of the macro unit to capture the microscopic characteristics of the beam structure; a plane model is established for the two-dimensional RVE, and a corresponding solid model is established for the three-dimensional RVE. The RVE is finite element meshed using corresponding two-dimensional or three-dimensional units to obtain a mesoscopic scale grid, which is used as the mesoscopic unit of the beam structure. During the meshing process, it is necessary to select an appropriate grid size and unit type to ensure the accuracy and reliability of the calculation results.

[0053] S03, then establish the periodic boundary conditions and equilibrium equations between the macroscopic and mesoscopic units. The periodic boundary conditions are shown in equation 1:

[0054] Formula 1:

[0055]

[0056] in, and Respectively represent the displacement vector of the right node of RVE and the corresponding displacement vector of the left node, and They represent the displacement vector of the upper node of RVE and the corresponding displacement vector of the lower node, L x and L y denote the length of RVE along the x and y directions, d i represents the node displacement vector of the macro unit, N i,x and N i,y They represent the derivatives of the macro unit shape function with respect to the x-coordinate and the y-coordinate respectively;

[0057] In multi-scale analysis, since the constitutive relation of the macroscopic unit cannot be directly obtained, the internal virtual work of the macroscopic unit can be converted into the internal virtual work of the mesoscopic unit according to the Hill-Mandel principle, thus establishing the Direct FE of the beam. 2 Multiscale finite element model;

[0058] Formula 2:

[0059] Among them, δW int is the internal virtual work of RVE, w α and J α are the weight and Jacobian at Gaussian integration point α, A is the cross-sectional area of ​​the beam, is the virtual strain of RVE, is the stress of RVE. In the process of establishing the equilibrium equation, the stiffness of RVE is scaled based on the Hill-Mandel principle, and the scaling coefficient is determined according to the following formula:

[0060]

[0061] where \(w\) is the scaling factor, and \(w\) α is the weight of the Gaussian integration point \(\alpha\) where the RVE is located, and \(J\) α is the value of the Jacobian determinant at the Gaussian integration point \(\alpha\) where the RVE is located, \(A\) is the cross-sectional area of the beam, and \(V\) α is the volume of the RVE at the Gaussian integration point \(\alpha\). The stiffness of the RVE can be scaled by scaling the volume of the RVE. For two-dimensional problems, the stiffness of the RVE can be scaled by magnifying the thickness of the RVE, and the magnification factor is \(w\). For three-dimensional problems, the RVE can be scaled proportionally in three directions to scale the stiffness of the RVE, and the magnification factor is

[0062] S04. Then, establish the objective function and volume constraint of topology optimization on the meso-elements and perform optimization. Taking maximizing the structural stiffness as an example, the objective function of the optimization is to minimize the strain energy of the structure, and the volume constraint is 60% of the initial volume. Here, the SIMP method is used for optimization;

[0063] The volume constraint is the percentage of the volume retained by the structure after optimization in the total volume of the structure before optimization. The purpose of the volume constraint is to limit the material usage of the beam structure to meet the cost and manufacturing constraints in practical engineering. The volume constraint is given by the designer before optimization according to the actual problem. After establishing the objective function and volume constraint, use a topology optimization algorithm, such as SIMP (Solid Isotropic Material with Penalization) or BESO (Bi-directional Evolutionary Structural Optimization), to optimize the beam structure. The topology optimization algorithm can automatically search for the optimal topology of the beam structure to meet the requirements of the objective function and volume constraint. Through optimization, the optimal meso-structure of the beam structure can be obtained to improve the stiffness, strength, and other performance indicators of the beam structure

[0064] S05. Finally, through iterative solution, obtain the optimized beam structure. It can be seen from the figure that the optimized beam structure meets the requirements of the volume constraint and has good stiffness.

[0065] The core of the present invention comprehensively considers the influence of the meso-scale of the inhomogeneous beam on the entire macro-scale, enabling the design of the inhomogeneous beam to not only consider the influence brought by material inhomogeneity but also save materials while ensuring the stiffness of the beam structure, reduce the calculation time of the finite element, and design a lighter variable cross-section beam structure, thereby improving the performance of the overall beam structure. To illustrate the structure, features, and effects of the present invention in detail, the following preferred examples are listed.

[0066] Example 1:

[0067] In this numerical example, the 2D Direct FE 2 The cantilever beam model is topologically optimized, such as Figure 5 As shown. The left end of the beam is completely fixed, and a vertical concentrated force F = 1N is applied to the right end of the beam. The geometric dimensions of the beam are: beam length L = 200mm, beam height h = 4mm, and beam thickness t = 1mm. The Young's modulus and Poisson's ratio of the beam are 7×10 4 MPa and 0.3. Direct FE 2 The cantilever beam model is discretized into 10 macro-scale elements, which are assigned a negligible Young's modulus, which is set to 1×10 -9 MPa. The dimensions of the RVE are b = 2 mm, h = 4 mm, and the thickness of the RVE is proportionally adjusted to 5 mm to meet energy consistency. Each RVE is discretized into 40*80 units. In order to maximize the stiffness of the cantilever beam, the optimization domain is all mesoscopic units, and the volume constraint fraction is 65%. Secondly, in order to ensure the continuity of the optimized structure, the units around each RVE are frozen.

[0068] Figure 6 It is a schematic diagram of the simulation results of the topology optimization design of a non-homogeneous cantilever beam with a variable cross-section under the scheme of the present invention. It can be seen from the figure that the design result has continuous variability, and its stiffness is similar to that of the traditional full finite element topology optimization result. The calculation time is saved by about 80% compared with the traditional single-scale topology optimization method. Figure 7 Schematic diagram of the traditional full finite element topology optimization results.

[0069] Example 2:

[0070] Figure 8 Direct FE 2 Simply supported beam model. The beam is subjected to a uniformly distributed force downward vertically, with a magnitude of q = 1N / m. The geometric dimensions of the beam are: beam length L = 1200mm, beam height h = 12mm, beam thickness t = 9mm, and the Young's modulus and Poisson's ratio of the beam are 7×10 4 MPa and 0.3. Due to the symmetry of the model, only half of the beam is modeled and discretized into five macroscale elements, which are given a negligible Young's modulus, which is set to 1×10 -9 MPa. The length of RVE is b = 12 mm, and the magnification factor of RVE is n = 1.710. In order to maximize the stiffness of the beam, the optimization domain is all mesoscopic elements, and the volume constraint fraction is 60%.

[0071] Fig. 9This is the topology optimization design result of the non-homogeneous beam with variable cross-section under the scheme of the present invention. It can be seen from the figure that the design result has continuous variability, and its stiffness is maximized under the given volume constraint. The calculation time saves about 92% of the time compared with the traditional single-scale topology optimization method. Fig.10 Schematic diagram of the traditional full finite element topology optimization results.

[0072] It should be noted that, in this article, the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, an element defined by the sentence "comprises a ..." does not exclude the existence of other identical elements in the process, method, article or device including the element.

[0073] The above are only preferred embodiments of the present invention, and are not intended to limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made using the contents of the present invention specification and drawings, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present invention.

Claims

1. A variable cross-section beam design method based on multi-scale finite element, characterized in that: The method comprises the following steps: S01, dividing the beam model into macro-scale grids, which are used as macro-units of the beam model; S02, establishing an RVE model at the integration point of the macro unit, and dividing the RVE into mesoscopic scale grids, which are used as the mesoscopic units of the beam structure; S03, then establish the periodic boundary conditions and equilibrium equations between the macroscopic and mesoscopic units; S04. Establishing the objective function and volume constraint of topology optimization on the mesoscopic unit so as to optimize the mesoscopic structure of the beam structure; S05. Finally, through multiple optimization iterations, the optimized beam structure is obtained; In step S03, the periodic boundary condition is as shown in Formula 1: Formula 1: ； in, and Respectively represent the displacement vector of the right node of RVE and the corresponding displacement vector of the left node, and Respectively represent the displacement vector of the upper node of RVE and the corresponding displacement vector of the lower node, and Respectively represent RVE along Direction and The length of the direction, represents the nodal displacement vector of the macro unit, and They represent the macro unit shape function pair Coordinates and Derivatives of coordinates; In multi-scale analysis, since the constitutive relation of the macroscopic unit cannot be directly obtained, the internal virtual work of the macroscopic unit can be converted into the internal virtual work of the mesoscopic unit according to the Hill-Mandel principle, thus establishing the Direct FE of the beam. 2 Multiscale finite element model; Formula 2: ； in, is the internal virtual work of RVE, and Gaussian integration points The weights and Jacobian at , is the cross-sectional area of ​​the beam, is the virtual strain of RVE, is the stress of RVE; In step S03, the stiffness of the RVE is scaled based on the Hill-Mandel principle during the process of establishing the equilibrium equation, and the scaling coefficient is determined according to the following formula: ； in is the scaling factor, is the Gaussian integration point where RVE is located The weight of is the Gaussian integration point where RVE is located The value of the Jacobian at , is the cross-sectional area of ​​the beam, Gaussian integration point The volume of the RVE at .

2. The variable cross-section beam design method based on multi-scale finite element according to claim 1, characterized in that: In step S01, the step of dividing the beam model into macro-scale grids to form macro-units of the beam model includes: First, the beam model is modeled, and the beam unit is used to mesh the beam model to obtain a macro-scale grid, which is used as the macro-unit of the beam model. After the beam unit is meshed, the corresponding boundary conditions need to be applied, including the fixed end, free end and load of the beam, to give the beam unit a Young's modulus that is negligible compared to the actual material.

3. The variable cross-section beam design method based on multi-scale finite element according to claim 1, characterized in that: In step S02, the steps of establishing an RVE model at the integration point of the macro unit and dividing the RVE into mesoscopic scale grids to use the grids as the mesoscopic units of the beam structure include: An RVE model is established at the integration point of the macro unit to capture the microscopic characteristics of the beam structure. A plane model is established for the two-dimensional RVE, and a corresponding solid model is established for the three-dimensional RVE. The RVE is divided into finite element grids using the corresponding two-dimensional or three-dimensional units to obtain a mesoscopic scale grid, and the actual material parameters, including Young's modulus and Poisson's ratio, are assigned to obtain the mesoscopic unit of the beam structure.

4. The variable cross-section beam design method based on multi-scale finite element according to claim 1, characterized in that: In step S04, before optimization, the objective function and volume constraint of topology optimization are established on the mesoscopic unit. When the structural stiffness is maximized: the objective function is the total strain energy of the structure, and the volume constraint is the percentage of the volume retained after structural optimization to the total volume of the structure before optimization; the purpose of the volume constraint is to limit the material usage of the beam structure to meet the cost and manufacturing constraints in actual engineering. After establishing the objective function and volume constraint, the topology optimization algorithm is used to optimize the beam structure.

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