Topological optimization method for simultaneously optimizing support position and structure topology

Abstract

The invention discloses a topological optimization method for simultaneously optimizing a support position and a structure topology, and relates to the technical field of topological optimization. The method specifically comprises the following steps: determining an initial structure design domain, a support design domain and boundary conditions of topological optimization; establishing a topological optimization model with the minimum flexibility by adopting a variable density method; calculating sensitivity information of the target function and the constraint; solving a structural design domain and a support design domain based on a gradient optimization method and a discrete particle swarm algorithm; and verifying the validity of the method through a numerical case. According to the method, the optimal support position can be effectively determined through the discrete particle swarm optimization; a structure topology optimization model is established by adopting a variable density method, so that density variables in a design domain can be efficiently processed.

Description

Technical Field

[0001] The invention relates to the technical field of structural topology optimization, and in particular to a topology optimization method for simultaneously optimizing support positions and structural topology. Background Art

[0002] Topology optimization technology has unparalleled potential in reducing structural weight and enhancing structural performance. Since its introduction, it has been widely and deeply applied in many fields such as aerospace, automobile, construction, bridge, and new material design.

[0003] Traditional topology optimization often operates on a design domain with pre-set support positions. However, in actual engineering problem scenarios, the support positions of the structure are not fixed, but variable. Moreover, with the change of the support position, the most ideal structural topology will also change accordingly. This change relationship is of great significance in actual engineering because it directly affects the overall performance and applicability of the structure. For example, in the aerospace field, changes in the support position of the aircraft structure may have a key impact on flight performance, load capacity, etc.; in the automotive field, changes in the support position of the body structure may involve multiple aspects of the car's handling, safety, and fuel economy. Therefore, accurately grasping and processing this dynamic relationship between the support position and the structural topology is crucial to achieving a more optimized engineering structure design. Summary of the invention

[0004] The purpose of the present invention is to propose a topology optimization method for simultaneously optimizing support positions and structural topology. The discrete particle swarm algorithm can effectively determine the optimal support position; the variable density method is used to establish a structural topology optimization model, which can efficiently process density variables in the design domain.

[0005] To achieve the above object, the present invention proposes a topology optimization method for simultaneously optimizing support positions 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, using variable density method to establish the topology optimization model with minimum compliance;

[0008] S3, calculating 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 S1, a topology optimization structure design domain Ω and a topology optimization support design domain φ are constructed, boundary conditions are determined, and the structural stiffness is maximized by optimizing the material distribution and the support unit positions in the design domain.

[0012] Preferably, in S2, when establishing the topology optimization model, the structural design domain and the support design domain are first meshed, and a density variable x is assigned to each mesh in the structural design domain, and its value range is 0≤x≤1; a discrete variable y is assigned to each mesh in the support design domain, and y=0 or 1; in order to avoid checkerboards in the structural design domain and reduce gray units, the density variable x is filtered and projected to obtain the physical density x corresponding to the unit e , the unit elastic modulus is obtained by physical density interpolation, and the calculation formula of the unit 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 are the elastic modulus of the solid element, and E min <<E0; p is the penalty factor;

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

[0016]

[0017] Among them, S i is the volume of the i-th support design domain unit; y i is the density of the i-th support unit; S * is the preset number of support units, and 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 is the unit displacement vector, k0 is the unit stiffness matrix under unit Young's modulus, N is the number of control support units, V(x) and V0 are the material volume and design domain volume respectively, f is the given volume fraction, v 0eis the volume of the unit design domain, e is the e-th unit in the design domain, x e is the physical density corresponding to the unit.

[0022] Preferably, in S3, in the topology optimization based on the gradient algorithm, the sensitivity information of the objective function and the constraints is necessary, and the derivative of the compliance objective function with respect to the physical density is expressed as:

[0023]

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

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

[0026] S41, solving the structural design domain based on the gradient optimization method;

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

[0028] S43, judging whether the convergence condition is met: if the change values ​​of all design variables meet the accuracy requirements or reach the preset maximum number of iteration steps, the iteration is stopped, otherwise it returns to S41.

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

[0030] Preferably, in S42, the discrete particle swarm optimization algorithm comprises the following specific steps:

[0031] S421, random initialization: the position z of each particle i (t), speed v i (t), i is the particle number, t is the current iteration number, and the fitness value f(z i (t)), where the fitness value is the flexibility value of the overall structure;

[0032] S422, update the best position of the particle: record the best position pbest of each particle in its history i , record the best of the historical best positions of all particles in the group as the global best position gbest i ;

[0033] S423. Update the speed and position of particles: The formula for updating the speed of each particle is as follows:

[0034]

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

[0036] Use the Sigmoid function to convert the speed value into a probability value between 0 and 1, and update the particle position according to this probability value. The formula is as follows:

[0037]

[0038] Among them, v i (t+1) is the probability value after conversion, j is the jth 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 reaches the preset maximum number of iterations, stop the iteration, otherwise return to step S422.

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

[0042] Therefore, the present invention proposes a topology optimization method for simultaneously optimizing support positions and structural topology, and its beneficial effects are as follows:

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

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

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

[0046] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a flow chart of a topology optimization method for simultaneously optimizing support positions and structural topology proposed by the present invention;

[0048] Figure 2 A topology optimization method for simultaneously optimizing support positions and structural topology proposed in the present invention is to perform mesh division on a cantilever beam structure;

[0049] Figure 3 It is a schematic diagram of an initial domain for cantilever beam topology optimization in a topology optimization method for simultaneously optimizing support positions and structural topology proposed by the present invention;

[0050] Figure 4 It is a schematic diagram of setting a symmetric boundary topology optimization initial domain for a cantilever beam in a topology optimization method for simultaneously optimizing support positions and structural topology proposed by the present invention;

[0051] Figure 5 It is a curve diagram showing the change of the flexibility of the cantilever beam structure with iteration in a topology optimization method for simultaneously optimizing the support position and the structural topology proposed by the present invention;

[0052] Figure 6 This is a comparison diagram of the flexibility of the cantilever beam after optimization for different support positions in a topology optimization method for simultaneously optimizing support positions and structural topology proposed in the present invention. DETAILED DESCRIPTION

[0053] In order to make the technical solutions, advantages and purposes of the present invention clearer, the technical solutions of the embodiments of the present invention are clearly and completely described below. The described embodiments are part of the embodiments of the present invention, rather than all the embodiments. Based on the described embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work belong to the protection scope of this application.

[0054] Unless otherwise defined, technical or scientific terms used in the present invention shall have the common meanings understood by one having ordinary skills in the field to which the present invention belongs.

[0055] like Figure 1 As shown, the present invention provides a topology optimization method for simultaneously optimizing support positions and structural topology, and the specific steps are as follows:

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

[0057] S2. Use the variable density method to establish a topology optimization model with minimum compliance. When establishing the topology optimization model, first mesh the structural design domain and the support design domain; Figure 2 As shown in the figure, a density variable x is assigned to each grid in the structural design domain, and its value range is 0≤x≤1; a discrete variable y is assigned to each grid in the support design domain, y=0 or 1; in order to avoid the checkerboard in the structural design domain and reduce the grayscale units, the density variable x is filtered and projected to obtain the physical density x corresponding to the unit e, the unit elastic modulus is obtained by physical density interpolation, and the calculation formula of the unit 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 are the elastic modulus of the solid element, and E min <<E0; p is the penalty factor;

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

[0061]

[0062] Among them, S i is the volume of the i-th support design domain unit; y i is the density of the i-th support unit; S * is the preset number of support units, and m is the number of support design domain units.

[0063] In summary, the topology optimization model formula 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 is the unit displacement vector, k0 is the unit stiffness matrix under unit Young's modulus, N is the number of control support units, V(x) and V0 are the material volume and design domain volume respectively, f is the given volume fraction, v 0e is the volume of the unit design domain, e is the e-th unit in the design domain, x e is the physical density corresponding to the unit.

[0067] S3. In topology optimization based on gradient algorithm, the sensitivity information of objective function and constraint is necessary. The derivative of compliance objective function to physical density is expressed as:

[0068]

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

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

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

[0072] 2. Solve the support design domain based on the discrete particle swarm optimization algorithm. The specific steps of the discrete particle swarm optimization algorithm are as follows:

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

[0074] 2.2. Update the best position of the particle: record the best position pbest of each particle in its history i , record the best of the historical best positions of all particles in the group as the global best position gbest i ;

[0075] 2.3. Update the speed and position of particles: The formula for updating the speed of each particle is as follows:

[0076]

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

[0078] Use the Sigmoid function to convert the speed value into a probability value between 0 and 1, and update the particle position according to this probability value. The formula is as follows:

[0079]

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

[0081] 2.4. Evaluate the new position: calculate the fitness value of each particle at the new position;

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

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

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

[0085] Example 1

[0086] The technical solution of this patent is further described in detail below in conjunction with specific embodiments.

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

[0088] like Figure 5 As shown in the figure, the iterative optimization history of the structure is given. As the structure becomes clearer, the iteration tends to be stable, and finally the support position is stabilized at 1 unit away from the endpoint. Due to the OC solution characteristics, the volume fraction constraint is always maintained at the set 0.5.

[0089] like Figure 6 As shown in the figure, the flexibility comparison of different support positions is shown. The first column is the optimal support position obtained by the particle swarm. It can be observed that the flexibility value of the first column is the smallest, which proves that this method is feasible. The symbol "+" represents that the support positions on the upper and lower sides are approaching the center at the same time, and "-" represents the opposite, which means that the support positions on the upper and lower sides are moving away from the center at the same time. When the difference from the optimal support position is one unit, the final structural flexibility value is not much different. As it is farther away from the optimal support position, the flexibility increases. Among them, Figure 6 The data in are shown in Table 1.

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

[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 for simultaneously optimizing support position and structural topology, and introduces support design domain and structural design domain at the same time. Through the discrete particle swarm algorithm, the optimal support position can be effectively determined; the variable density method is used to establish a structural topology optimization model, which can efficiently process the density variables in the design domain.

[0093] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A topology optimization method for simultaneously optimizing support positions and structural topology, characterized in that: Here are the steps: S1. Determine the initial structural design domain, support design domain and boundary conditions for topology optimization; S2, using variable density method to establish the topology optimization model with minimum compliance; S3, calculating 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.

2. A topology optimization method for simultaneously optimizing support positions and structural topology according to claim 1, characterized in that: In S1, a topology optimization structural design domain Ω and a topology optimization support design domain φ are constructed, boundary conditions are determined, and the structural stiffness is maximized by optimizing the material distribution and the support unit position in the design domain.

3. A topology optimization method for simultaneously optimizing support positions and structural topology according to claim 1, characterized in that: In S2, when establishing the topology optimization model, the structural design domain and the support design domain are first meshed, and a density variable x is assigned to each mesh in the structural design domain; a discrete variable y is assigned to each mesh in the support design domain; in order to avoid checkerboards in the structural design domain and reduce grayscale units, the density variable x is filtered and projected to obtain the physical density x corresponding to the unit. e , the unit elastic modulus is obtained by physical density interpolation, and the calculation formula of the unit elastic modulus is as follows: E e (x e )=E min +x e p (E0-E min ); Among them, E0 and E min are the elastic modulus of the solid element, and E min <<E0; p is the penalty factor; The number of support units is determined by constraining the support design domain volume. The specific constraint formula is as follows: Among them, S i is the volume of the i-th support design domain unit; y i is the density of the i-th support unit; S * is the preset number of support units, and m is the number of support design domain units.

4. A topology optimization method for simultaneously optimizing support positions and structural topology according to claim 3, characterized in that: The topology optimization model formula is as follows: Ku=F; 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 is the unit displacement vector, k0 is the unit stiffness matrix under unit Young's modulus, N is the number of control support units, V(x) and V0 are the material volume and design domain volume respectively, f is the given volume fraction, v 0e is the volume of the unit design domain, e is the e-th unit in the design domain, x e is the physical density corresponding to the unit.

5. The topology optimization method for simultaneously optimizing support positions and structural topology according to claim 1, characterized in that: In S3, in the topology optimization based on the gradient algorithm, the sensitivity information of the objective function and the constraints is necessary, and the derivative of the compliance objective function to the physical density is expressed as: Since the physical density is obtained by filtering and projecting the density variable, for different filtering and projection functions, based on the chain derivation criterion, the sensitivity of the objective function to the density variable is finally obtained: Where c is the compliance objective function.

6. A topology optimization method for simultaneously optimizing support positions and structural topology according to claim 1, characterized in that: In S4, the structural design domain and the support design domain are solved based on the gradient optimization method and the discrete particle swarm algorithm. The specific calculation steps are as follows: S41, solving the structural design domain based on the gradient optimization method; S42, solving the support design domain based on discrete particle swarm optimization algorithm; S43, judging whether the convergence condition is met: if the change values ​​of all design variables meet the accuracy requirements or reach the preset maximum number of iteration steps, the iteration is stopped, otherwise it returns to S41.

7. A topology optimization method for simultaneously optimizing support positions and structural topology according to claim 6, characterized in that: In S41, a gradient-based OC optimization algorithm is used to solve the structural design domain.

8. A topology optimization method for simultaneously optimizing support positions and structural topology according to claim 6, characterized in that: In S42, the discrete particle swarm optimization algorithm specifically performs the following steps: S421, random initialization: the position z of each particle i (t), speed x i (t), i is the particle number, t is the current iteration number, and the fitness value f(z i (t)), where the fitness value is the flexibility value of the overall structure; S422, update the best position of the particle: record the best position pbest of each particle in its history i , record the best of the historical best positions of all particles in the group as the global best position gbest i ; S423. Update the speed and position of particles: The formula for updating the speed of each particle is as follows: Among them, w is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are random numbers; Use the Sigmoid function to convert the speed value into a probability value between 0 and 1, and update the particle position according to this probability value. The formula is as follows: Among them, v i (t+1) is the probability value after conversion, j is the jth support unit in the i-th particle; 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 reaches the preset maximum number of iterations, stop the iteration, otherwise return to step S422.

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

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