A game theory based structural multi-material topology optimization method and system

Abstract

The application relates to the technical field of structural design and optimization, and discloses a structural multi-material topology optimization method and system based on game theory. The method first determines an initial finite element structure according to engineering needs, and constructs a topology optimization model taking a phase field continuous density as a design variable. A complete information dynamic game method is used to determine the optimization sequence of multi-material, and the multi-material optimization problem is converted into a series of two-material optimization problems of solid material and empty material. Compared with the prior art, the application uses the idea of game theory for multi-material topology optimization, simplifies the multi-material optimization process, and obtains various structural forms, thereby providing engineers with various choices for selecting structural forms according to various constraint conditions. The application can solve the problems that the existing several types of multi-material optimization methods in the prior art are complex in concept and not easy to implement, and the optimization variables involved are too many, thereby causing a huge calculation amount.

Description

Technical Field

[0001] This invention relates to the field of structural design and optimization technology, and in particular to a method and system for multi-material topology optimization of structures based on game theory. Background Technology

[0002] Engineering structures are often composed of more than one material. Therefore, it is crucial to consider the use of dual or even multiple materials in optimization design. This not only leverages the advantages of various material properties but also yields better results than single-material structures. However, existing multi-material optimization methods are either conceptually complex and difficult to implement, or involve too many optimization variables, resulting in enormous computational costs. Therefore, proposing a multi-material optimization method that is easy to implement and has fewer optimization variables has become an urgent technical problem to be solved. Summary of the Invention

[0003] This invention provides a multi-material topology optimization method and system based on game theory to solve the problems of existing multi-material optimization methods being conceptually complex and difficult to implement, and involving too many optimization variables leading to huge computational loads.

[0004] To achieve the above objectives, the present invention employs the following technical solution:

[0005] In a first aspect, the present invention provides a multi-material topology optimization method based on game theory, comprising:

[0006] S1: Determine the initial finite element structure according to the engineering requirements, and construct a topology optimization model with the phase field continuity density as the design variable based on the initial finite element structure;

[0007] S2: The optimization order of multiple materials is determined by a dynamic game of complete information. Based on the optimization order, the multi-material optimization problem is transformed into a series of dual-material optimization problems involving solid materials and empty materials. The dual-material optimization problems are solved sequentially. During the optimization process, the sum of the two materials in each cell remains unchanged.

[0008] S3: This concludes one iteration. Proceed to the next iteration until the iteration conditions are met, then exit the iteration process to obtain the optimized topology.

[0009] Secondly, this application provides a multi-material topology optimization system based on game theory, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the method described in the first aspect above.

[0010] Beneficial effects:

[0011] This invention provides a game theory-based multi-material topology optimization method. First, an initial finite element structure is determined according to engineering needs, and a topology optimization model is constructed with the phase field continuity density as the design variable. Then, a complete information dynamic game method is used to determine the optimization order of the multiple materials, transforming the multi-material optimization problem into a series of dual-material optimization problems involving both solid and empty materials. Compared to existing technologies, this invention utilizes game theory for multi-material topology optimization, which not only simplifies the multi-material optimization process but also yields diverse structural forms, providing engineers with a variety of choices based on various constraints. It solves the problems of existing multi-material optimization methods being conceptually complex and difficult to implement, and involving too many optimization variables leading to enormous computational costs. Attached Figure Description

[0012] Figure 1 This is a flowchart of a preferred embodiment of the present invention for a multi-material topology optimization method based on game theory;

[0013] Figure 2 This is the initial finite element structure diagram of the first topology to be optimized according to a preferred embodiment of the present invention;

[0014] Figure 3 This is an optimization result diagram of the first topology to be optimized in a preferred embodiment of the present invention;

[0015] Figure 4 This is the initial finite element structure diagram of the second topology to be optimized in a preferred embodiment of the present invention;

[0016] Figure 5 This is an optimization result diagram of the second topology to be optimized in a preferred embodiment of the present invention;

[0017] Figure 6 This is a schematic diagram of the node connections of the initial finite element structure in a preferred embodiment of the present invention. Detailed Implementation

[0018] The technical solution of the present invention will be clearly and completely described below. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Similarly, the terms "an" or "a" and similar terms do not indicate a quantity limitation, but rather indicate the presence of at least one. The terms "connected" or "linked" and similar terms are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. "Up," "down," "left," "right," etc., are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship also changes accordingly.

[0020] Please see Figure 1 This application provides a game theory-based method for topology optimization of multi-material structures, comprising:

[0021] S1: Determine the initial finite element structure according to the engineering requirements, and construct a topology optimization model with the phase field continuity density as the design variable based on the initial finite element structure;

[0022] S2: The optimization order of multiple materials is determined by a dynamic game of complete information. Based on the optimization order, the multi-material optimization problem is transformed into a series of dual-material optimization problems involving solid materials and empty materials. The dual-material optimization problems are solved sequentially. During the optimization process, the sum of the two materials in each cell remains unchanged.

[0023] S3: This concludes one iteration. Proceed to the next iteration until the iteration conditions are met, then exit the iteration process to obtain the optimized topology.

[0024] The aforementioned game theory-based multi-material topology optimization method first determines the initial finite element structure according to engineering needs, and constructs a topology optimization model with the phase field continuity density as the design variable. It then employs a full-information dynamic game theory method to determine the optimization order of the multiple materials, transforming the multi-material optimization problem into a series of dual-material optimization problems involving both solid and empty materials. Compared to existing technologies, this invention utilizes game theory for multi-material topology optimization, which not only simplifies the multi-material optimization process but also yields diverse structural forms, providing engineers with a variety of choices based on various constraints. This addresses the problems of existing multi-material optimization methods being conceptually complex and difficult to implement, and involving too many optimization variables leading to enormous computational costs.

[0025] The steps of the above-described multi-material topology optimization method based on game theory are described in detail below using a complete example:

[0026] 1. Determine the initial design of the target based on the project requirements, and set the boundary conditions of the structure according to the design requirements.

[0027] 2. Topology optimization with the goal of minimizing structural strain energy. Strain energy is defined here as the sum of work done by external forces, and can be expressed as:

[0028] C = F T U;

[0029] Where F is the external force load array; U is the displacement array.

[0030] 3. Based on the initial finite element structure of the topology, construct a topology optimization model with the continuous density of the phase field as the design variable, the physical model based on the phase field method, and the strain energy as the objective.

[0031] Among them, the multi-material topology optimization model considering game theory is:

[0032]

[0033] Among them, R N×m Let g(x) be a real matrix, x be the density variable; f(x) be the objective function, F be the external load matrix, and U be the displacement matrix; g(x) be a real matrix. h ) represents the volume constraint of the h-th material, N is the total number of elements, i is the element number, and v i It is the volume of the i-th unit. It is the design volume of the h-th material. is the relative density of the h-th material in the i-th unit, with a value ranging from 0 to 1, and m is the type of material being considered.

[0034] 4. First, sort the materials, placing the empty material last. In each iteration, perform dual-material optimization with the first material and the empty material, followed by dual-material optimization with the second material and the empty material. The objective for each solid material is to minimize structural flexibility, until all non-empty materials are optimized with the empty material. The objective function for each material in each iteration is as follows:

[0035] Material 1: min C = F T U(x1);

[0036] Material 2: min C = F T U(x2);

[0037] Where U(x1) and U(x2) represent that when optimizing the distribution of material 1 or material 2, the displacement response of the structure is only related to the corresponding material distribution.

[0038] 5. In each dual-material optimization, the classic OC optimizer is used for solution:

[0039]

[0040] Where, move is the upper limit of change in a single iteration, η is the damping coefficient, and B is the fixed-point iterative expression constructed by the optimality condition, (x k ) h Let λ represent the design variable vector for the h-th material in the k-th iteration. h The Lagrange multiplier for the volume constraint of the h-th material is generally determined by the dichotomy method;

[0041] and The sensitivities of the objective function and the volume constraint of the h-th material, respectively, satisfy the following relationship:

[0042]

[0043] in, To optimize the displacement of element i when the h-th material distribution is used, k h Let k be the element stiffness matrix of the h-th material. m Here, we define the element stiffness matrix for the m-th material, i.e., the empty material, and set the minimum elastic modulus E for the empty material. min =10 -9 To prevent singularity.

[0044] 6. Stop this iteration and start the next iteration until the maximum number of iterations is met or the maximum change value of the design variable is less than the given threshold.

[0045] In this embodiment, the optimal topology scheme is to obtain a topology structure with optimized strain energy after removing a given volume, given structural material properties, structural optimization region, constraints, load type, load magnitude, load location, and moving boundary point location and type.

[0046] In this embodiment, a physical model based on the phase-field method with continuous phase-field variables is employed. This method assumes that all elements of the topology are composed of various materials with relative densities between 0 and 1, using the relative density of each material in each element as the design variable. In all examples, the topology is constructed using isotropic materials with elastic moduli E1 = 2.0, E2 = 1.0, and Poisson's ratio of 0.3. The optimization objective is to minimize the structural strain energy while retaining 50% of the volume.

[0047] Furthermore, specific examples are provided using MBB beams and cantilever beam structures. In the finite element analysis, the elements are discretized into quadrilateral elements of 120×40, 180×60, 240×80 and 80×50, 120×75, 160×100, 240×150 respectively. Their respective loads and initial boundary conditions are as follows: Figure 2 and Figure 4As shown. The results of the example are provided in Figure 3 and Figure 5 The node connections of the initial finite element structure are as follows: Figure 6 As shown. Since the MBB beam is a symmetrical structure, half of it is used for optimization design. Material 1 is represented in black, material 2 in gray, and empty materials in white.

[0048] In summary, the multi-material optimization concept based on game theory presented in this application is simple and easy to implement, and the number of optimization variables is the same as that of traditional two-material optimization. Compared with existing topology optimization techniques, it can achieve a better shape, thus realizing a further extension based on existing topology optimization methods.

[0049] In the preferred method, the idea of ​​dynamic game with complete information in game theory is used to transform multi-material optimization into a series of non-empty material and empty material optimizations, which can reduce the number of optimization variables. At the same time, the concept is simple and easy to implement.

[0050] This application also provides a game theory-based multi-material topology optimization system, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the above-described method. This synthetic gradient natural potential observation system can implement various embodiments of the above-described synthetic gradient natural potential observation method and achieve the same beneficial effects; further details are omitted here.

[0051] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A multi-material topology optimization method based on game theory, characterized in that, include: S1: Determine the initial finite element structure according to the engineering requirements, and construct a topology optimization model with the phase field continuity density as the design variable based on the initial finite element structure; S2: The optimization order of multiple materials is determined by a dynamic game of complete information. Based on the optimization order, the multi-material optimization problem is transformed into a series of dual-material optimization problems involving solid materials and empty materials. The dual-material optimization problems are solved sequentially. During the optimization process, the sum of the two materials in each cell remains unchanged. S3: This concludes one iteration. Proceed to the next iteration until the iteration conditions are met, then exit the iteration process to obtain the optimized topology. S1 includes: The initial design of the target is determined according to the project requirements, and the boundary conditions of the structure are set according to the design requirements. Based on the initial finite element structure of the topology, a topology optimization model is constructed with the phase field continuous density as the design variable, the phase field method as the physical model, and the minimization of structural strain energy as the optimization objective. The strain energy represents the sum of work done by external forces. The topology optimization model satisfies the following relationship: ； in, Represents a real matrix. Represents density variables; Describe the objective function. It is an array of external loads. It is a displacement matrix; This represents the volume constraint of the h-th material. It is the total number of units. For unit number, It is the first Volume of each unit It is the design volume of the h-th material. It is the first The relative density of the h-th material in each unit ranges from 0 to 1, where m is the type of material being considered, and T represents transpose. When solving the dual-material optimization problem, the OC optimizer is used, satisfying the following relationship: ； in, This represents the upper limit of change in a single iteration. Let B be the damping coefficient, and let B be the fixed-point iterative expression constructed based on the optimality conditions. Let h be the Lagrange multiplier for the volume constraint of the h-th material. This represents the design variable vector for the h-th material in the k-th iteration; The process of ensuring that the sum of the two materials remains constant within each unit during optimization includes: Before each dual-material optimization, the sum of the relative densities of the two materials in each cell is determined and kept constant during the optimization process; the objective function for each material in each iteration is as follows: Material 1: ; Material 2: ; Where C represents the sum of work done by external forces. , This means that when optimizing the distribution of material 1 or material 2, the displacement response of the structure is only related to the corresponding material distribution.

2. The multi-material topology optimization method based on game theory according to claim 1, characterized in that, The process of transforming the multi-material optimization problem into a series of dual-material optimization problems involving solid materials and empty materials based on the optimization order includes: Based on the idea of ​​dynamic game with complete information, each non-empty material continuously adjusts its composition to achieve optimal structural performance. The materials are sorted in descending order according to the size of the elastic modulus, with the empty material placed last. Then, each non-empty material and the empty material are optimized in turn.

3. The multi-material topology optimization method based on game theory according to claim 1, characterized in that, The sequential solution of the dual-material optimization problem includes: In each iteration, dual-material optimization is first performed with the first material and the empty material, followed by dual-material optimization with the second material and the empty material. The optimization objective for each solid material is to minimize the structural flexibility, until all non-empty materials are dual-material optimized with the empty material.

4. The multi-material topology optimization method based on game theory according to claim 1, characterized in that, The unit in S2 is represented by a phase-field model, and the sum of the relative density or volume fraction of all materials in each unit is 1.

5. The multi-material topology optimization method based on game theory according to claim 1, characterized in that, The iteration conditions include: satisfying the maximum number of iterations or satisfying the specified cell density convergence condition.

6. A multi-material topology optimization system based on game theory, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 5.

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