A topology optimization method of supporting position and structure topology simultaneous optimization

Abstract

The application discloses a kind of topological optimization methods of support position and structure topology simultaneous optimization, and relates to the technical field of topological optimization.The method specifically includes: determining the initial structure design domain of topological optimization, support design domain and boundary condition;Using variable density method to establish the topology optimization model of minimum compliance;The sensitivity information of objective function and constraint is calculated;Based on gradient optimization method and discrete particle swarm algorithm, structure design domain and support design domain are solved;The effectiveness of the method is verified by numerical case.The present application can effectively determine the optimal support position by discrete particle swarm algorithm;Using variable density method to establish structure topology optimization model, density variables in design domain can be efficiently processed.

Description

Technical Field

[0001] This invention relates to the field of structural topology optimization technology, and in particular to a topology optimization method that simultaneously optimizes support locations and structural topology. Background Technology

[0002] Topology optimization technology possesses unparalleled potential in reducing structural weight and enhancing structural performance. Since its introduction, this technology has been widely and deeply applied in many fields such as aerospace, automotive, architecture, bridges, and new materials design.

[0003] Traditional topology optimization often operates within a design domain with predefined support locations. However, in real-world engineering problems, the support locations of structures are not static but variable. Furthermore, as the support locations change, the ideal structural topology also changes accordingly. This dynamic relationship is significant in practical engineering because it directly impacts the overall performance and applicability of the structure. For example, in the aerospace field, changes in the support locations of aircraft structures can have critical effects on flight performance and load-bearing capacity; in the automotive field, changes in the support locations of the vehicle body structure can affect various aspects of vehicle performance, including handling, safety, and fuel economy. Therefore, accurately grasping and managing this dynamic relationship between support locations and structural topology is crucial for achieving more optimized engineering structural designs. Summary of the Invention

[0004] The purpose of this invention is to propose a topology optimization method that simultaneously optimizes support location and structural topology. By using the discrete particle swarm optimization algorithm, the optimal support location can be effectively determined. By using the variable density method to establish a structural topology optimization model, the density variables in the design domain can be handled efficiently.

[0005] To achieve the above objectives, this invention proposes a topology optimization method that simultaneously optimizes support location and structural topology, the steps of which are as follows:

[0006] S1. Determine the initial structural design domain, support design domain, and boundary conditions for topology optimization;

[0007] S2. A topology optimization model with minimum compliance is established using the variable density method.

[0008] S3. Calculate the sensitivity information of the objective function and constraints;

[0009] S4. Solve the structural design domain and support design domain based on gradient optimization method and discrete particle swarm optimization algorithm;

[0010] S5. Verify the effectiveness of the method.

[0011] Preferably, in step S1, a topology-optimized structural design domain Ω and a topology-optimized support design domain φ are constructed, boundary conditions are determined, and the structural stiffness is maximized by optimizing the material distribution and support unit positions in the design domains.

[0012] Preferably, in step S2, when establishing the topology optimization model, the structural design domain and the support design domain are first divided into meshes. A density variable x is assigned to each mesh in the structural design domain, with a value range of 0 ≤ x ≤ 1. A discrete variable y is assigned to each mesh in the support design domain, where y = 0 or 1. To avoid checkerboard patterns in the structural design domain and to reduce grayscale units, the density variable x is filtered and projected to obtain the physical density x corresponding to the unit. e The element elastic modulus is obtained by interpolation of the physical density, and the formula for calculating the element elastic modulus is as follows:

[0013] E e (x e ) = E min +x e p (E0-E min );

[0014] Among them, E0 and E min Let E be the elastic modulus of the solid element, and E be the elastic modulus of the solid element. min <<E0; p is the penalty factor;

[0015] The number of support elements is determined by constraining the volume of the support design domain. The specific constraint formula is as follows:

[0016]

[0017] Among them, S i Let y be the volume of the i-th supporting design domain element; i S represents the density of the i-th support unit; * The preset number of support units is denoted by m, where m is the number of support design domain units.

[0018] Preferably, the topology optimization model formula is as follows:

[0019]

[0020] Ku = F;

[0021] Where c is the structural compliance, u and F are the global displacement and force vectors, K is the global stiffness matrix, and u e Let v be the element displacement vector, k0 be the element stiffness matrix at unit Young's modulus, N be the number of control support elements, V(x) and V0 be the material volume and design domain volume, respectively, f be the given volume fraction, and v 0eLet x be the volume of the design domain, e be the e-th element in the design domain, and x be the volume of the design domain. e This represents the physical density corresponding to the unit.

[0022] Preferably, in step S3, the sensitivity information of the objective function and constraints is essential in the gradient-based topology optimization. The derivative of the compliance objective function with respect to the physical density is expressed as:

[0023]

[0024] Since physical density is obtained by filtering and projecting density variables, for different filtering and projection functions, based on the chain rule, the sensitivity of the objective function to the density variable can be finally obtained. Where c is the compliance objective function.

[0025] Preferably, in step S4, the structural design domain and support design domain are solved based on the gradient optimization method and the discrete particle swarm optimization algorithm. The specific calculation steps are as follows:

[0026] S41. Solving the structural design domain based on gradient optimization method;

[0027] S42. Solving the support design domain based on discrete particle swarm optimization algorithm;

[0028] S43. Determine if the convergence condition has been met: If the change values ​​of all design variables reach the accuracy requirements or the preset maximum number of iterations, stop the iteration; otherwise, return to S41.

[0029] Preferably, in step S41, the gradient-based OC optimization algorithm is used to solve the structural design domain.

[0030] Preferably, in step S42, the specific steps of the discrete particle swarm optimization algorithm are as follows:

[0031] S421. Random initialization: the position z of each particle i (t), velocity v i (t), where i is the particle number and t is the current iteration step, calculate the fitness value f(z) for each particle. i (t)), where the fitness value is the overall compliance value of the structure;

[0032] S422, Update the best position of particles: Record the best position pbest in the history of each particle. i The best position among all the historical best positions of the particles in the swarm is recorded as the global best position, gbest. i ;

[0033] S423, Update particle velocity and position: The formula for updating the velocity of each particle is as follows:

[0034]

[0035] Where w is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are random numbers;

[0036] The sigmoid function is used to convert the velocity value into a probability value between 0 and 1. The particle position is then updated based on this probability value, as shown in the following formula:

[0037]

[0038] Among them, v i (t+1) is the converted probability value, and j is the j-th support unit in the i-th particle;

[0039] S424. Evaluate the new position: Calculate the fitness value of each particle at the new position;

[0040] S425. Check the termination condition: If the fitness value of each particle is equal or the preset maximum number of iterations is reached, stop the iteration; otherwise, return to step S422.

[0041] Preferably, in step S5, the topology optimization result of the optimal support is obtained by implementing numerical cases, and it is compared with the results of other support positions; the effectiveness of the method is verified by comparing the compliance values.

[0042] Therefore, this invention proposes a topology optimization method that simultaneously optimizes support location and structural topology, with the following beneficial effects:

[0043] (1) The present invention introduces both the support design domain and the structural design domain, which can simultaneously optimize the structural topology and support position;

[0044] (2) The present invention uses discrete particle swarm optimization algorithm to optimize the support design domain, which can obtain a clear support structure;

[0045] (3) The present invention uses the variable density method (SIMP method) to establish a topology optimization model and adopts a gradient-based criterion method, which can efficiently optimize the structural design domain.

[0046] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0047] Figure 1 This is a flowchart of a topology optimization method that simultaneously optimizes support location and structural topology, as proposed in this invention.

[0048] Figure 2 This invention proposes a topology optimization method that simultaneously optimizes support location and structural topology, specifically for mesh generation of cantilever beam structures.

[0049] Figure 3 This is a schematic diagram of the initial domain for topology optimization of a cantilever beam in a topology optimization method that simultaneously optimizes support position and structural topology proposed in this invention.

[0050] Figure 4 This is a schematic diagram of the initial domain for topology optimization of a cantilever beam, which is set as a symmetrical boundary in a topology optimization method that simultaneously optimizes the support position and structural topology proposed in this invention.

[0051] Figure 5 This is a graph showing the variation of the cantilever beam structure's compliance with iterations in a topology optimization method that simultaneously optimizes support position and structural topology, as proposed in this invention.

[0052] Figure 6 This is a comparison of the compliance of a cantilever beam after optimization for different support positions in a topology optimization method that simultaneously optimizes support position and structural topology proposed in this invention. Detailed Implementation

[0053] To make the technical solutions, advantages, and objectives of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below. The described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the protection scope of this application.

[0054] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0055] like Figure 1 As shown, this invention provides a topology optimization method that simultaneously optimizes support location and structural topology. The specific steps are as follows:

[0056] S1. Determine the topology optimization structural design domain Ω and the topology optimization support design domain φ, determine the boundary conditions, and maximize the structural stiffness by optimizing the material distribution and support unit positions in the design domain.

[0057] S2. A topology optimization model with minimum compliance is established using the variable density method. When establishing the topology optimization model, the structural design domain and the support design domain are first meshed; for example... Figure 2 As shown, a density variable x is assigned to each grid in the structural design domain, with a value range of 0 ≤ x ≤ 1; a discrete variable y is assigned to each grid in the support design domain, where y = 0 or 1; to avoid checkerboard patterns in the structural design domain and to reduce grayscale elements, the density variable x is filtered and projected to obtain the physical density x corresponding to the element. eThe element elastic modulus is obtained by interpolation of the physical density, and the formula for calculating the element elastic modulus is as follows:

[0058] E e (x e ) = E min +x e p (E0-E min );

[0059] Among them, E0 and E min Let E be the elastic modulus of the solid element, and E be the elastic modulus of the solid element. min <<E0; p is the penalty factor;

[0060] The number of support elements is determined by constraining the volume of the support design domain. The specific constraint formula is as follows:

[0061]

[0062] Among them, S i Let y be the volume of the i-th supporting design domain element; i S represents the density of the i-th support unit; * The preset number of support units is denoted by m, where m is the number of support design domain units.

[0063] In summary, the formula for the topology optimization model is as follows:

[0064]

[0065] Ku = F;

[0066] Where c is the structural compliance, u and F are the global displacement and force vectors, K is the global stiffness matrix, and u e Let v be the element displacement vector, k0 be the element stiffness matrix at unit Young's modulus, N be the number of control support elements, V(x) and V0 be the material volume and design domain volume, respectively, f be the given volume fraction, and v 0e Let x be the volume of the design domain, e be the e-th element in the design domain, and x be the volume of the design domain. e This represents the physical density corresponding to the unit.

[0067] S3. In gradient-based topology optimization, sensitivity information of the objective function and constraints is essential. The derivative of the compliance objective function with respect to the physical density is expressed as:

[0068]

[0069] Since physical density is obtained by filtering and projecting density variables, for different filtering and projection functions, based on the chain rule, the sensitivity of the objective function to the density variable can be finally obtained. Where c is the compliance objective function.

[0070] S4. Solve the structural design domain and support design domain based on gradient optimization and discrete particle swarm optimization. The specific calculation steps are as follows:

[0071] 1. The gradient-based OC optimization algorithm is used to solve the structural design domain;

[0072] 2. Solving the supporting design domain based on the Discrete Particle Swarm Optimization (DPSO) algorithm. The specific steps of the DPSO algorithm are as follows:

[0073] 2.1 Random initialization: The position z of each particle i (t), velocity v i (t), where i is the particle number and t is the current iteration step, calculate the fitness value f(z) for each particle. i (t));

[0074] 2.2 Update the best position of particles: Record the best position (pbest) in the history of each particle. i The best position among all the historical best positions of the particles in the swarm is recorded as the global best position, gbest. i ;

[0075] 2.3 Updating Particle Velocity and Position: The formula for updating the velocity of each particle is as follows:

[0076]

[0077] Where w is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are random numbers;

[0078] The sigmoid function is used to convert the velocity value into a probability value between 0 and 1. The particle position is then updated based on this probability value, as shown in the following formula:

[0079]

[0080] Among them, v i (t+1) is the converted probability value, and j is the j-th support unit in the i-th particle;

[0081] 2.4 Evaluate the new location: Calculate the fitness value of each particle at the new location;

[0082] 2.5 Check the termination condition: Stop iteration if the fitness value of each particle is equal or the preset maximum number of iterations is reached; otherwise, return to step 2.2.

[0083] 3. Determine if the convergence condition has been met: If the change values ​​of all design variables reach the accuracy requirements or the preset maximum number of iterations, stop the iteration; otherwise, return to S41.

[0084] S5. By implementing numerical cases, the topology optimization results of the optimal support are obtained and compared with the results of other support locations; the effectiveness of the method is verified by comparing the compliance values.

[0085] Example 1

[0086] The technical solution of this patent will be further described in detail below with reference to specific embodiments.

[0087] like Figure 3 As shown, in this embodiment, a classic cantilever beam is used. To reduce computational complexity, symmetrical boundary conditions are applied, and half of the structure is optimized. Figure 4 As shown, the design domain and support design domain sizes are set to 80mm × 25mm and 1mm × 25mm, respectively. The force F is set to 50N, with the force direction being vertically downward. The mesh size and shell thickness are set to 1mm, Young's modulus E = 100Gpa, Poisson's ratio μ = 0.3, the design domain volume fraction is set to 0.5, the penalty exponent p = 3, the filter radius is set to 3, and the number of support elements is 1. The objective is to minimize compliance and find the optimal support location.

[0088] like Figure 5 As shown, the history of structural iterative optimization is presented. As the structure gradually becomes clearer, the iteration tends to stabilize, and the final support position is stabilized at 1 unit away from the endpoint. Due to the OC solution characteristics, the volume fraction constraint has always been maintained at the set 0.5.

[0089] like Figure 6 As shown, the diagram compares the compliance of different support positions. The first column represents the optimal support positions obtained by particle swarm optimization. Observation reveals that the first column has the lowest compliance value, proving the feasibility of this method. The symbols "+" indicate that the support positions on both the top and bottom edges move towards the center simultaneously, while "-" indicates the opposite, meaning that the support positions on both the top and bottom edges move away from the center simultaneously. For positions differing from the optimal support position by one unit, the final structural compliance value is not significantly different; the compliance increases with distance from the optimal support position. Figure 6 The data is shown in Table 1.

[0090] Table 1 Comparison of compliance at different support locations

[0091] Support position Softness c / J Optimal support 2.3168 -1 2.3730 +1 2.3875 +2 2.5195 +3 2.6614 +6 3.2705 +9 4.0978

[0092] Therefore, the present invention proposes a topology optimization method that simultaneously optimizes support location and structural topology. It introduces both support design domain and structural design domain, and can effectively determine the optimal support location through discrete particle swarm optimization algorithm. The method of variable density is used to establish a structural topology optimization model, which can efficiently handle the density variables in the design domain.

[0093] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A topology optimization method that simultaneously optimizes support location and structural topology, characterized in that, The steps are as follows: S1. Determine the initial structural design domain, support design domain, and boundary conditions for topology optimization; S2. A topology optimization model with minimum compliance is established using the variable density method. S3. Calculate the sensitivity information of the objective function and constraints; S4. Solve the structural design domain and support design domain based on gradient optimization method and discrete particle swarm optimization algorithm; S5. Verify the effectiveness of the method; In step S2, when establishing the topology optimization model, the structural design domain and the support design domain are first divided into meshes, and a density variable is assigned to each mesh in the structural design domain. ; Assign discrete variables to each grid supporting the design domain. To avoid checkerboard patterns in the structural design domain and reduce grayscale units, the density variable... By performing filtering and projection, the physical density corresponding to the unit is obtained. The element elastic modulus is obtained by interpolation of the physical density, and the formula for calculating the element elastic modulus is as follows: ； in, and These are the elastic moduli of the solid element, and they have... ; As a penalty factor; The number of support elements is determined by constraining the volume of the support design domain. The specific constraint formula is as follows: ； in, For the first The volume of each supporting design domain unit; For the first The density of each support unit; To preset the number of support units, To support the number of design domain units; In S3, the sensitivity information of the objective function and constraints is essential in the gradient-based topology optimization. The derivative of the compliance objective function with respect to the physical density is expressed as: ； Since physical density is obtained by filtering and projecting density variables, for different filtering and projection functions, based on the chain rule, the sensitivity of the objective function to the density variable can be finally obtained. ,in, Let be the compliance objective function. For element displacement vectors, The element stiffness matrix is ​​given by unit Young's modulus. This represents the physical density corresponding to the unit.

2. The topology optimization method for simultaneously optimizing support location and structural topology according to claim 1, characterized in that, In S1, a topology optimization structure design domain is constructed. Topology optimization supports design domain By determining the boundary conditions and optimizing the material distribution and support unit positions in the design domain, the structural stiffness can be maximized.

3. The topology optimization method for simultaneously optimizing support location and structural topology according to claim 1, characterized in that, The formula for the topology optimization model is as follows: ； ； ； in, For structural compliance, and For global displacement and force vectors, The global stiffness matrix, For element displacement vectors, The element stiffness matrix is ​​given by unit Young's modulus. To control the number of support units, and These are the material volume and the design domain volume, respectively. For a given volume fraction, The volume of the design domain for the unit. For the first in the design domain Units, This represents the physical density corresponding to the unit.

4. The topology optimization method for simultaneously optimizing support location and structural topology according to claim 1, characterized in that, In step S4, the structural design domain and support design domain are solved based on the gradient optimization method and the discrete particle swarm optimization algorithm. The specific calculation steps are as follows: S41. Solving the structural design domain based on gradient optimization method; S42. Solving the support design domain based on discrete particle swarm optimization algorithm; S43. Determine if the convergence condition has been met: If the change values ​​of all design variables reach the accuracy requirements or the preset maximum number of iterations, stop the iteration; otherwise, return to S41.

5. The topology optimization method for simultaneously optimizing support location and structural topology according to claim 4, characterized in that, In step S41, the gradient-based OC optimization algorithm is used to solve the structural design domain.

6. The topology optimization method for simultaneously optimizing support location and structural topology according to claim 4, characterized in that, In step S42, the specific steps of the discrete particle swarm optimization algorithm are as follows: S421. Random initialization: the position of each particle ,speed , Number the particles. Calculate the fitness value for each particle at the current iteration step. The fitness value is the overall structural compliance value; S422. Update the best position of particles: Record the best position of each particle in its history. Record the best position among all the historical best positions of all particles in the swarm as the global best position. ; S423, Update particle velocity and position: The formula for updating the velocity of each particle is as follows: ； in, It is inertial weight. It is the acceleration constant. It is a random number; The sigmoid function is used to convert the velocity value into a probability value between 0 and 1. The particle position is then updated based on this probability value, as shown in the following formula: ； ； in, The converted probability value. For the first The first particle One support unit; S424. Evaluate the new position: Calculate the fitness value of each particle at the new position; S425. Check the termination condition: If the fitness value of each particle is equal or the preset maximum number of iterations is reached, stop the iteration; otherwise, return to step S422.

7. The topology optimization method for simultaneously optimizing support location and structural topology according to claim 1, characterized in that, In step S5, the topology optimization result of the optimal support is obtained by implementing numerical cases and compared with the results of other support positions; the effectiveness of the method is verified by comparing the compliance values.

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