A topology optimization structural design method based on spatial control points

Abstract

The application provides a topology optimization structure design method based on a space control point, and a space point control technology for structure topology optimization. The SPCT constructs a field function by using the space arrangement of the control points, effectively reduces the number of design variables of the topology optimization, improves the optimization calculation efficiency, and avoids the appearance of the chessboard effect and the medium density elements in the traditional topology optimization method. The control point placement method is intuitive and convenient for controlling the initial configuration or sample data, and further improves the optimization efficiency. In addition, the application derives the sensitivity information of the SPCT topology optimization model, establishes a gradient-based optimization framework, and can use a non-gradient algorithm to solve the scheme without the aid of gradient information to solve complex nonlinear problems. The technology has obvious advantages in reducing design variables and improving calculation efficiency, and has broad application prospects in the field of structure optimization design.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of structural optimization design, and aims at the demand of reducing the dimension of design variables, and proposes a topology optimization design method based on spatial control points. BACKGROUND

[0002] In traditional topology optimization, the number of design variables is huge and the same as the number of finite element meshes in structure division, and usually only gradient-based optimization algorithms can be used for solving. This not only seriously limits the efficiency of optimization solving, but also hinders the application of topology optimization algorithms in multidisciplinary and highly nonlinear structural design problems. Some topology optimization methods based on level set or moving components can reduce the number of design variables to some extent, but still retain at least hundreds of design variables and completely rely on gradient algorithms for solving. MFSE reduces a large number of design variables, but due to the superposition characteristics of the field function, samples and initial solutions cannot be added arbitrarily, and it is difficult to deal with the improvement problem of known structures. In order to break through this bottleneck.

[0003] The prior art is as follows: a method, device, medium and equipment for topology optimization of a double-layer flexible thermoelectric device are disclosed in publication No. CN118568799B, a structure non-gradient topology optimization method based on a sequence Kriging proxy model is disclosed in publication No. CN110852011B, and the comprehensive results are that the control points and the field function make the topology optimization model free from the dependence on the number of meshes, reduce the number of design variables, mapping is a necessary mathematical tool for the results to approach 0 and 1 distribution (which is required for gradient-based and non-gradient-based optimization algorithms for solving), and sensitivity derivation is for the use of gradient-based optimization algorithms for solving. SUMMARY

[0004] The application proposes a topology optimization structural design method based on spatial control points. The method constructs a spatial field function through control points in space, and converts the field function into a topology structure by means of a mapping function. Since the spatial control points are directly used as design variables, the number of design variables in the optimization model is greatly reduced. This not only significantly improves the optimization solving efficiency based on gradient algorithms, but also creates conditions for the application of non-gradient optimization algorithms, thereby enhancing the universality of topology optimization in multidisciplinary problems, and the technical scheme is as follows:

[0005] A topology optimization structural design method based on spatial control points, comprising spatial field function establishment, topology optimization model establishment and sensitivity analysis, characterized by comprising the following steps:

[0006] Step 1: constructing a field function based on spatial control point technology to reduce the number of design variables: arranging control points in space, constructing a field function through the spatial arrangement of control points, wherein the specific value of the control point is used as a design variable;

[0007] Step 2: Establish a topology optimization model, and map the field function to the structure topology through a mapping function;

[0008] Step 3: Sensitivity analysis is performed to provide solution conditions for gradient-based optimization algorithms: the sensitivity information of the objective function and the volume constraint function with respect to the design variables is derived.

[0009] The application also discloses a non-volatile storage medium, characterized in that the non-volatile storage medium comprises a stored program, wherein the program controls a device in which the non-volatile storage medium is located to execute the method when the program is executed.

[0010] The application also discloses a terminal device, characterized in that the terminal device comprises a processor, a memory, a communication interface and a bus; the processor, the memory and the communication interface are connected through the bus and complete communication among each other; the memory stores executable program code; the processor runs a program corresponding to the executable program code by reading the executable program code stored in the memory, so as to execute the method.

[0011] Advantages

[0012] The spatial control point technology breaks through the dependence of the topology optimization model on the number of grids, and can significantly reduce the number of design variables. This improvement not only greatly improves the optimization solving efficiency, so that it can easily deal with large-scale finite element grid topology optimization problems, but also gives the spatial control point high arrangement flexibility, which can be arbitrarily configured, and is convenient for modifying known samples and initial structures, so that the design scheme is more flexible and diverse. The significant reduction of the design variables enables the topology optimization problem to be solved based on a non-gradient algorithm, and can be applied to multi-field coupling problems, significantly enhancing the versatility of topology optimization. In addition, thanks to the spatial continuity and correlation of the field function, this technology can effectively avoid the checkerboard effect in traditional topology optimization, and with the help of the mapping technology, it can also avoid the appearance of gray elements.

[0013] The application proposes a spatial point control technology (SPCT) for structure topology optimization. SPCT uses the spatial arrangement of control points to construct a field function, effectively reducing the number of design variables in topology optimization, improving optimization calculation efficiency, and avoiding the appearance of checkerboard effect and medium density elements. The control point placement method is intuitive and convenient for controlling the initial configuration or sample data, further improving the optimization efficiency. In addition, the application derives the sensitivity information of the topology optimization model based on SPCT, establishes a gradient-based optimization framework, and provides a non-gradient solution scheme to solve complex nonlinear problems. This technology has obvious advantages in reducing design variables and improving calculation efficiency, and has broad application prospects in the field of non-gradient optimization. BRIEF DESCRIPTION OF DRAWINGS

[0014] Figure 1 Schematic diagram of spatial control points and field functions;

[0015] Figure 2 Schematic diagram of the field function mapping structure;

[0016] Figure 3 Optimization problem diagram;

[0017] Figure 4 Schematic diagrams of topology optimization results under different control point layouts; where: (a) shows the field function distribution and control point distribution of the optimization results when the control point interval is 4mm; (b) shows the topology optimization results when the control point interval is 4mm; (c) shows the field function distribution and control point distribution of the optimization results when the control point interval is 6mm; (d) shows the topology optimization results when the control point interval is 6mm; (e) shows the field function distribution and control point distribution of the optimization results when the control point interval is 8mm; (f) shows the topology optimization results when the control point interval is 8mm; (g) shows the field function distribution and control point distribution of the optimization results when the control point interval is 10mm; (h) shows the topology optimization results when the control point interval is 10mm.

[0018] Figure 5 A schematic diagram of the iterative process. Detailed Implementation

[0019] This invention proposes a Spatial Points Control Technology (SPCT) for structural topology optimization. SPCT utilizes the spatial arrangement of control points to construct field functions, effectively reducing the number of design variables in topology optimization, improving computational efficiency, and avoiding the chessboard effect and the occurrence of medium-density elements. Its control point placement method is intuitive, facilitating control of initial configurations or sample data, further improving optimization efficiency. Furthermore, this invention derives sensitivity information based on the SPCT topology optimization model, establishes a gradient-based optimization framework, and provides a non-gradient solution scheme to solve complex nonlinear problems. This technology has significant advantages in reducing design variables and improving computational efficiency, and has broad application prospects, especially in the field of non-gradient optimization.

[0020] The image optimization issues that need to be addressed include: Figure 3 As shown, the minimum flexibility of the structure is designed, and a 40% volume constraint is retained. As an example, the Poisson's ratio in the material parameters is set to 0.3, the elastic modulus is set to 1 Pa, the external load is 1 N, and a simply supported constraint is applied. A total of 7200 finite element meshes are generated.

[0021] The first step is to construct the field function based on spatial control point technology.

[0022] 1.1) In the two-dimensional design domain space, control points are arranged at intervals of 4mm, 6mm, 8mm, and 10mm, resulting in four working conditions with 496, 231, 128, and 91 control points respectively. By changing their arrangement in the vertical direction, field functions of different shapes can be constructed.

[0023] 1.2) Subsequently, a specific field function can be established by controlling the distribution of these points. Among existing mathematical tools, the Kriging model is a field function model that exhibits spatial correlation and effectively meets the usage requirements. Therefore, we directly use the Kriging model to construct the field function, the specific expression of which is as follows:

[0024]

[0025] in, Indicates at any point The field function value at point Z represents the values ​​of Ns control points, which are used to control the shape of the field; I is a vector whose elements are all 1 and whose number of elements is the same as the number of control points. It is an intermediate quantity, determined by the second formula in the curly braces according to the Kriging method theory; It is the correlation vector between the control point and any point in space, where R is the correlation matrix between control points, and its specific form is as follows:

[0026]

[0027] in, Represents the coordinates of Ns control points. R and The elements in the array are represented as follows:

[0028]

[0029] Here, n represents the dimension of the topology optimization problem, which is 2 in this example, and k takes the value of... i and j are used to distinguish different control points in space, i.e. Point and The absolute value of the difference between the k-th dimension coordinates; It is the correlation coefficient of the k-th dimension, since The magnitude of this directly affects distance-based spatial correlation, and is controlled using the following formula. :

[0030] because The magnitude of this value directly affects distance-based spatial correlation, therefore careful consideration of its value is crucial. Therefore, the following formula is used to control... :

[0031]

[0032] Here, A parameter representing the correlation between a single, adjustable control point, while ensuring that the correlation between control points is no longer affected by the distance between them. The impact.

[0033] The second step is to establish a topology optimization model.

[0034] 2.1) Introduce a mapping function to map the field function to the structure. The schematic diagram of the mapping result is shown below. Figure 2 The formula is as follows:

[0035]

[0036] in It is the relative density used in the topology optimization after mapping, where e is the natural constant. It is a parameter that controls the degree of mapping; during the optimization process, Gradually increasing the value from 0.2 to 2 can increase the relative density. By gradually approximating the 0-1 distribution, the final topology optimization result is obtained. During the optimization iteration process, when the change in the optimization objective is less than 5%, α is increased by 0.1 or 1.1 times, whichever is smaller, and is determined by the following formula:

[0037]

[0038] 2.2) Based on the above method, for the topology optimization problem of continuum structures, taking the minimum compliance problem as an example, the expression of the topology optimization model of the Space Point Control Technique (SPCT) is as follows:

[0039]

[0040] in, It is the minimum compliance objective function. These are nodal forces in the finite element method. These are nodal displacements, which can be used to analyze the stress and strain information of a structure. It includes the overall stiffness matrix, the structure's elastic modulus, Poisson's ratio, and other parametric properties. This is the current volume fraction. It is a volume constraint.

[0041] The third step is sensitivity analysis to prepare a solution using a gradient-based optimization algorithm. This third step is unnecessary if a non-gradient algorithm is used. In this example, a gradient-based optimization algorithm is used.

[0042] 3.1) Analyze the sensitivity of the objective function. For ease of analysis, the objective function is expanded into the following form:

[0043]

[0044] Where N is the number of finite element meshes for the structure. The element stiffness matrix contains material property information used to assemble the overall stiffness matrix. is the element displacement vector, used to assemble the nodal displacement vectors; P is the penalty coefficient; It is the first Coordinates of the center of each finite element; objective function about The derivative form is as follows:

[0045]

[0046] and The specific form is as follows:

[0047]

[0048]

[0049] in, , for The i-th element; according to the chain rule, the above derivation can be obtained as follows: All information; including For the objective function with respect to the i-th design variable The derivative of .

[0050] 3.2) Analyze the sensitivity of the volume constraint function. The expanded form of the volume constraint function is as follows:

[0051]

[0052] Its derivative relationship is:

[0053]

[0054] The constraints of the optimization model and the sensitivity of the objective function have been derived, and optimization can be performed. A gradient-based optimization algorithm was used to obtain the material distribution (optimal topology) in the design domain. The design results are as follows: Figure 4 As shown, the iterative process using the gradient-based MMA algorithm is as follows: Figure 5 As shown, the detailed records of the optimized data are recorded in the following table:

[0055]

[0056] This invention aims to significantly improve the computational efficiency and versatility of topology optimization. By cleverly utilizing spatial control points to construct a topology optimization method, it effectively solves the problems of high dimensionality of design variables and low optimization efficiency in traditional methods. This method is compatible with gradient and non-gradient optimization algorithms, flexibly adapting to different engineering scenarios, and provides an efficient and universal solution for the field of engineering structure optimization design.

[0057] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention. The scope of protection claimed by the appended claims and their equivalents is defined.

Claims

1. A topology optimization structure design method based on spatial control points, comprising three parts: establishing a spatial field function, establishing a topology optimization model, and sensitivity analysis, characterized by: Includes the following steps: Step 1: Constructing a field function based on spatial control point technology to reduce the number of design variables: Arrange control points in space, and construct a field function through the spatial arrangement of control points, where specific values ​​of the control points are used as design variables; Step 2: Establish a topology optimization model, and map the field function to the structural topology through a mapping function: Step 3: Perform sensitivity analysis to provide solution conditions for gradient-based optimization algorithms: derive the sensitivity information of the objective function and volume constraint function with respect to design variables; Step 1 includes the following: using the Kriging model to construct the field function, the specific expression of which is as follows: ； in, Indicates at any point The field function value at point Z represents the values ​​of Ns control points, which are used to control the shape of the field; I is a vector whose elements are all 1 and whose number of elements is the same as the number of control points. It is an intermediate quantity, determined by the second formula in the curly braces according to the Kriging method theory; It is the correlation vector between the control point and any point in space, where R is the correlation matrix between control points, and its specific form is as follows: ； in, Represents the coordinates of Ns control points; R and The elements in the middle are represented as follows: ； Here, n represents the dimension of the topology optimization problem, and k takes the value of i and j are used to distinguish different control points in space, i.e. Point and The absolute value of the difference between the k-th dimension coordinates; It is the correlation coefficient of the k-th dimension, since The magnitude of this directly affects distance-based spatial correlation, and is controlled using the following formula. : ； Here, A parameter representing the correlation between a single, adjustable control point can be used to directly measure the correlation between control points. Regulation; For the topology optimization problem of continuum structures, taking the minimum compliance problem as an example, the expression of the spatial point control topology optimization model is as follows: ； in, It is the minimum compliance objective function. These are nodal forces in the finite element method. These are nodal displacements, which can be used to analyze the stress and strain information of a structure. It is the overall stiffness matrix, which includes the structure's elastic modulus and Poisson's ratio. This is the current volume fraction. It is a volume constraint.

2. The topology optimization structure design method based on spatial control points according to claim 1, characterized in that: step 2 includes the following: the mapping function formula is as follows: ； in It is the relative density used in the topology optimization after mapping, where e is the natural constant. It is a parameter that controls the degree of mapping; during the optimization process, Gradually increasing the value from 0.2 to 2 can increase the relative density. By gradually approximating the 0-1 distribution, the final topology optimization result is obtained. During the optimization iteration process, when the change in the optimization objective is less than 5%, α is increased by 0.1 or 1.1 times, whichever is smaller, and is determined by the following formula: ； in, It is newly generated The value, the mapping function, is to make the relative density approach the distribution of 0 and 1, so as to obtain the topology optimization result with a clear outline.

3. The topology optimization structure design method based on spatial control points according to claim 1, characterized in that: the objective function is expanded into the following form: ； in, N is the number of finite element meshes generated for the structure. The element stiffness matrix contains material property information used to assemble the overall stiffness matrix. is the element displacement vector, used to assemble the nodal displacement vectors; P is the penalty coefficient; It is the first Coordinates of the center of each finite element; objective function about The derivative form is as follows: ； and The specific form is as follows: ； ； in, , for The i-th element; according to the chain rule, the above derivation can be obtained as follows: All information; including For the objective function with respect to the i-th design variable The derivative of .

4. The topology optimization structure design method based on spatial control points according to claim 1, characterized in that: the sensitivity of the volume constraint function is analyzed, and the expansion form of the volume constraint function is as follows: ； Its derivative relationship is: ； The constraints of the optimization model and the sensitivity of the objective function have been derived, and optimization can be performed to obtain the optimal topology configuration in the design domain. Alternatively, if a non-gradient optimization algorithm is used, the topology optimization model can be solved directly without sensitivity analysis.

5. A non-volatile storage medium, characterized in that, The non-volatile storage medium includes a stored program, wherein the program, when executed, controls the device where the non-volatile storage medium is located to perform the method described in any one of claims 1 to 4.

6. A terminal device, characterized in that, The terminal device includes: a processor, a memory, a communication interface, and a bus; the processor, the memory, and the communication interface are connected through the bus and communicate with each other; the memory stores executable program code; the processor reads the executable program code stored in the memory to run a program corresponding to the executable program code, so as to execute the method as described in any one of claims 1-4 above.

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