Topological optimization structure design method based on space control points

Abstract

The invention provides a topological optimization structure design method based on space control points and a space point control technology of structure topological optimization. The SPCT constructs a field function by utilizing spatial arrangement of control points, so that the number of design variables of topological optimization is effectively reduced, the optimization calculation efficiency is improved, and a chessboard effect and medium-density elements in a traditional topological optimization method are avoided. The control point placement mode is visual, the initial configuration or sample data can be conveniently controlled, and the optimization efficiency is further improved. Besides, sensitivity information based on an SPCT topological optimization model is deduced, a gradient-based optimization framework is established, and a non-gradient algorithm solution scheme can be used to solve the complex nonlinear problem without the help of gradient information. The technology has obvious advantages in the aspects of reducing design variables and improving calculation efficiency, and particularly has wide application prospects in the field of structure optimization design.

Description

Technical Field

[0001] The present invention belongs to the field of structural optimization design. Aiming at the demand of reducing the dimension of design variables, a topology optimization design method based on spatial control points is proposed. Background Art

[0002] In traditional topology optimization, the number of design variables is as large as the number of finite element meshes used to divide the structure, and solutions can usually only be obtained by relying on gradient-based optimization algorithms. This not only severely limits the efficiency of the optimization solution, but also hinders the application of topology optimization algorithms in multidisciplinary and highly nonlinear structural design problems. Some current topology optimization methods based on level sets or moving components can reduce the number of design variables to a certain extent, but they still retain at least hundreds of design variables and rely entirely on gradient algorithms for solutions. MFSE reduces the number of design variables significantly, but due to the superposition characteristics of field functions, samples and initial solutions cannot be added arbitrarily, making it difficult to address the improvement of known structures. In order to break through this bottleneck, we need to use the existing topology optimization methods.

[0003] The existing technologies are as follows: Announcement No.: CN 118568799B discloses a method, apparatus, medium and equipment for topology optimization of a double-layer flexible thermoelectric device; Announcement No.: CN110852011 B discloses a structural non-gradient topology optimization method based on a sequential Kriging proxy model, etc. The comprehensive results, control points and field functions free the topology optimization model from dependence on the number of grids, reducing the number of design variables. Mapping is a necessary mathematical tool to make the results approach the distribution of 0 and 1 (required for both gradient-based and non-gradient solution algorithms), and sensitivity derivation is used to solve using a gradient-based optimization algorithm. Summary of the Invention

[0004] This paper proposes a topology optimization structural design method based on spatial control points. This method constructs a spatial field function using control points in space and transforms the field function into a topological structure using a mapping function. Because the spatial control points are directly used as design variables, the number of design variables in the optimization model is significantly reduced. This not only significantly improves the efficiency of gradient-based optimization solutions but also facilitates the application of non-gradient optimization algorithms, thereby enhancing the versatility of topology optimization in multidisciplinary problems. The technical solution is as follows:

[0005] A topology optimization structural design method based on spatial control points includes three parts: establishing a spatial field function, establishing a topology optimization model, and sensitivity analysis. The method is characterized by including the following steps:

[0006] Step 1: Construct 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 the specific values of the control points serve as design variables;

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

[0008] Step 3: Perform sensitivity analysis to provide solution conditions for the gradient-based optimization algorithm: derive the sensitivity information of the objective function and volume constraint function with respect to the design variables.

[0009] The present invention also discloses a non-volatile storage medium, characterized in that the non-volatile storage medium includes a stored program, wherein when the program is run, it controls the device where the non-volatile storage medium is located to execute the above method.

[0010] The present invention also discloses 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 complete communication with 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 above-mentioned method.

[0011] Beneficial effects

[0012] The spatial control point technology breaks through the topology optimization model's dependence on the number of meshes and can significantly reduce the number of design variables. This improvement not only greatly improves the efficiency of the optimization solution, making it easy to deal with topology optimization problems of large-scale finite element meshes, but also gives the spatial control points a high degree of layout flexibility. They can be arbitrarily configured, making it easy to modify known samples and initial structures, making the design scheme more flexible and diverse. The significant reduction in design variables allows 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. At the same time, with the help of mapping technology, it can also avoid the appearance of grayscale units.

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

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

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

[0016] Figure 3 Schematic diagram of the optimization problem;

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

[0018] Figure 5 Schematic diagram of the iterative process. DETAILED DESCRIPTION

[0019] The present invention proposes a spatial point control technology (SPCT) for structural topology optimization. SPCT uses the spatial arrangement of control points to construct field functions, effectively reducing the number of design variables for topology optimization, improving optimization calculation efficiency, and avoiding the appearance of checkerboard effects and medium-density elements. The control point placement method is intuitive, making it easy to control the initial configuration or sample data, further improving optimization efficiency. In addition, the present invention derives sensitivity information based on the SPCT topology optimization model, establishes a gradient-based optimization framework, and provides a non-gradient solution to solve complex nonlinear problems. This technology has obvious advantages in reducing design variables and improving calculation efficiency, and has broad application prospects, especially in the field of non-gradient optimization.

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

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

[0022] 1.1) In the two-dimensional design domain, control points are arranged at intervals of 4 mm, 6 mm, 8 mm, and 10 mm, resulting in four working conditions with 496, 231, 128, and 91 control points, respectively. By varying their vertical arrangement, different field functions 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 requirements of use. Therefore, we directly use the Kriging model to construct the field function, and its specific expression is as follows:

[0024]

[0025] in, Indicates that at any point The elements of the vector Z represent the values of the Ns control points, which are used to control the shape of the field. I is a vector whose elements are all 1 and the number of elements is the same as the number of control points. is an intermediate quantity, determined by the second formula in the brackets according to the Kriging method theory; is the correlation vector between the control point and any point in the 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 are represented as follows:

[0028]

[0029] Here, n represents the dimension of the topology optimization problem, which is 2 in this example, and k is , i and j are used to distinguish different control points in space, that is Indicates a point and The absolute value of the difference between the coordinates in the kth dimension; is the correlation coefficient of the kth dimension, since The size of directly affects the distance-based spatial correlation and is controlled by the following formula :

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

[0031]

[0032] Here, Represents a parameter that can independently adjust the correlation between control points, while making the correlation between control points no longer affected by the distance between them. impact.

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

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

[0035]

[0036] in is the relative density used in the topology optimization after mapping, e is a natural constant, is a parameter that controls the degree of mapping; during the optimization process, Gradually increasing from 0.2 to 2 can make the relative density Gradually approaching the 0-1 distribution, the final topology optimization result is obtained; during the optimization iteration process, when the change of each optimization target 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, the topology optimization model expression of the spatial point control technology (SPCT) is as follows, taking the minimum flexibility problem as an example:

[0039]

[0040] in, is the minimum flexibility objective function, is the nodal force in the finite element problem, is the node displacement, which can be used to analyze the stress and strain information of the structure. It is the overall stiffness matrix, which includes the elastic modulus, Poisson's ratio and other parameter properties of the structure. is the current volume fraction, is the volume constraint.

[0041] The third step is sensitivity analysis to solve the problem using a gradient-based optimization algorithm. If a non-gradient algorithm is used, the third step is not necessary. In this example, a gradient-based optimization algorithm is used.

[0042] 3.1) Analyze the sensitivity of the objective function. To facilitate analysis, expand the objective function into the following form:

[0043]

[0044] Where N is the number of finite element meshes divided into the structure, The element stiffness matrix contains material property information and is used to assemble the overall stiffness matrix. is the unit displacement vector, which is used to assemble the node displacement vector; P is the penalty coefficient; It is The coordinates of the center of the finite element; the objective function about The derivative form of is as follows:

[0045]

[0046] and The specific form is as follows:

[0047]

[0048]

[0049] in, , for The i-th element in; According to the chain rule, we can get All information of is the objective function for 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 can be optimized. The material distribution (optimal topology) in the design domain is obtained using a gradient-based optimization algorithm. The design results are as follows: Figure 4 As shown, the iterative process of the gradient-based MMA algorithm is as follows Figure 5 As shown, the details of the optimization 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 addresses the high dimensionality of design variables and low optimization efficiency inherent in traditional methods. This method is compatible with both gradient and non-gradient optimization algorithms, flexibly adapting to diverse engineering scenarios and providing an efficient and versatile solution for the optimization design of engineering structures.

[0057] The above shows and describes 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 above embodiments and descriptions merely illustrate the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed by the present invention is defined by the appended claims and their equivalents.

Claims

1. A topology optimization structural design method based on spatial control points, including three parts: establishing spatial field function, establishing topology optimization model, and sensitivity analysis. Its characteristics are: The steps include: Step 1: Construct 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 the specific values of the control points serve as design variables; Step 2: Establish a topology optimization model and map the field function to the structural topology through the mapping function: Step 3: Perform sensitivity analysis to provide solution conditions for the gradient-based optimization algorithm: derive the sensitivity information of the objective function and volume constraint function with respect to the design variables.

2. The topology optimization structural design method based on spatial control points according to claim 1 is characterized in that: step 1 includes the following content: using the Kriging model to construct a field function, the specific expression of which is as follows: ； in, Indicates that at any point The elements of vector Z represent 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 the number of elements is the same as the number of control points. is an intermediate quantity, determined by the second formula in the brackets according to the Kriging method theory; is the correlation vector between the control point and any point in the 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 are represented as follows: ； Here, n represents the dimension of the topology optimization problem, and the value of k is , i and j are used to distinguish different control points in space, that is Indicates a point and The absolute value of the difference between the coordinates in the kth dimension; is the correlation coefficient of the kth dimension, since The size of directly affects the distance-based spatial correlation and is controlled by the following formula : ； Here, Represents a parameter that can adjust the correlation between control points separately, so that the correlation between control points can be directly Regulation.

3. The topology optimization structural design method based on spatial control points according to claim 1, wherein step 2 comprises the following: the mapping function formula is as follows: ； in is the relative density used in the topology optimization after mapping, e is a natural constant, is a parameter that controls the degree of mapping; during the optimization process, Gradually increasing from 0.2 to 2 can make the relative density Gradually approaching the 0-1 distribution, the final topology optimization result is obtained; during the optimization iteration process, when the change of each optimization target 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 mapping function is used to make the relative density approach the distribution of 0 and 1, so as to obtain a topology optimization result with a clear outline.

4. The topology optimization structural design method based on spatial control points according to claim 1 is characterized in that: for the topology optimization problem of a continuum structure, the topology optimization model expression of the spatial point control, taking the minimum flexibility problem as an example, is as follows: ； in, is the minimum flexibility objective function, is the nodal force in the finite element problem, is the node displacement, which can be used to analyze the stress and strain information of the structure. is the overall stiffness matrix, which includes the elastic modulus and Poisson's ratio parameter properties of the structure. is the current volume fraction, is the volume constraint.

5. The topology optimization structural design method based on spatial control points according to claim 1 is characterized in that the objective function is expanded into the following form: ； in, N is the number of finite element meshes divided into the structure, The element stiffness matrix contains material property information and is used to assemble the overall stiffness matrix. is the unit displacement vector, which is used to assemble the node displacement vector; P is the penalty coefficient; It is The coordinates of the center of the finite element; the objective function about The derivative form of is as follows: ； and The specific form is as follows: ； ； in, , for The i-th element in; According to the chain rule, we can get All information of is the objective function for the i-th design variable The derivative of .

6. The topology optimization structural design method based on spatial control points according to claim 1 is characterized by analyzing the sensitivity of the volume constraint function, wherein the expanded 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 the optimization solution can be performed to obtain the optimal topological configuration in the design domain. In addition, if a non-gradient optimization algorithm is used for the solution, the topology optimization model can be solved directly without the need for sensitivity analysis.

7. A non-volatile storage medium, characterized in that: The non-volatile storage medium includes a stored program, wherein when the program is executed, the device where the non-volatile storage medium is located is controlled to execute the method according to any one of claims 1 to 6.

8. 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 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 as described in any one of claims 1 to 6 above.

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