A multi-material microstructure topology optimization method and system based on a double-network PINN

Abstract

The application belongs to the technical field of the intersection of computational mechanics and artificial intelligence, and discloses a multi-material microstructure topology optimization method and system based on a double-network PINN. The method comprises the following steps: establishing a material distribution network and a displacement field network; obtaining a local elastic tensor field and a local thermal expansion coefficient field in a design domain by using multi-phase material distribution weights and displacement field calculation; constructing a physical loss function and an objective loss function by using the local elastic tensor field and the local thermal expansion coefficient field; establishing a total loss function about the volume integral beam constraint, the objective loss function and the physical loss function, adjusting network parameters of the material distribution network and the displacement field network to minimize the total loss function, and then obtaining multi-phase material distribution weights at the minimum loss function, so as to realize topology optimization of the microstructure. Through the application, the problem that elastic performance and thermal expansion coefficient performance cannot be unified in optimization design is solved.

Description

Technical Field

[0001] This invention belongs to the technical field of computational mechanics and artificial intelligence, and more specifically, relates to a method and system for topology optimization of multi-material microstructures based on dual-network PINN. Background Technology

[0002] Existing microstructure design methods mainly include topology optimization methods based on finite element method (FE) discretization and design methods based on parametric configurations. FE-based topology optimization methods, such as SIMP and BESO, can search for optimal structures under given constraints. However, these methods typically rely on high-resolution mesh discretization, and the computational scale and solution cost increase significantly with increasing design degrees of freedom. Furthermore, problems such as grayscale elements and jagged boundaries are prone to occur during the updating of discrete variables. While empirical parametric design methods are simple to implement, their design space is limited, making it difficult to fully exploit the performance potential of complex microstructures.

[0003] In recent years, neural network methods have been applied to materials design and structural optimization. These methods can represent complex configurations using continuous functions, reducing reliance on explicit discrete meshes to some extent. However, purely data-driven methods often lack necessary physical constraints, and the generated structures may not meet mechanical equilibrium conditions, leading to insufficient physical reliability of the design results. Physical Information Neural Networks (PINNs) provide a new technical approach for structural response prediction by incorporating the residuals of partial differential equations into the training process. However, most existing PINN methods are used for forward problem solving, i.e., predicting the physical field response under given material distribution conditions. Research directly applying them to topology optimization, especially in the reverse design of multi-material microstructures, remains limited.

[0004] Furthermore, existing methods mostly focus on optimization design under single material or single performance objective conditions, and do not adequately support the synergistic design of elastic and thermal expansion properties in multi-material microstructures. In particular, in multi-material design, the elastic modulus, thermal expansion coefficient, and spatial distribution of different material phases jointly determine the final macroscopic response, and existing technologies struggle to simultaneously achieve unified optimization among continuous representation of multi-phase materials, physical field constraint embedding, and homogenization performance objectives.

[0005] Therefore, there is an urgent need in the existing technology for a topology optimization method for multi-material microstructures, which can simultaneously introduce mechanical equilibrium constraints and combine homogenization performance objectives on the basis of continuous characterization of material distribution, so as to achieve synergistic optimization design of elastic performance and thermal expansion performance. Summary of the Invention

[0006] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a multi-material microstructure topology optimization method and system based on dual-network PINN, solving the problem of the inability to uniformly optimize the design of elastic properties and thermal expansion coefficients.

[0007] To achieve the above objectives, according to one aspect of the present invention, a topology optimization method for multi-material microstructures based on dual-network PINN is provided, the method comprising the following steps: A material distribution network is established, with the coordinates of each point in the design domain of the microstructure to be optimized as the output and the multiphase material distribution weights of the microstructure to be optimized as the input; at the same time, a displacement field network is established, with the multiphase material distribution weights of the microstructure to be optimized as the input and the microscopic displacement fields at various points on the microstructure to be optimized as the output. The local elastic tensor field and local thermal expansion coefficient field within the design domain are calculated using the obtained multiphase material distribution weights and displacement field. The strong form residuals at various points within the design domain are calculated using the local elastic tensor field and local thermal expansion coefficient field, and the physical loss function of the design domain is constructed using the strong form residuals. At the same time, the target loss function is constructed using the local elastic tensor field and local thermal expansion coefficient field. A total loss function is established with respect to volume fraction constraints, objective loss function, and physical loss function. The network parameters of the material distribution network and displacement field network are adjusted to minimize the total loss function, thereby obtaining the multiphase material distribution weights at the minimum loss function, thus achieving topology optimization of the microstructure.

[0008] More preferably, the formula for the total loss function is as follows:

[0009] in, The target loss function is constructed based on the homogenization performance. This is the physical loss function corresponding to the residuals of the mechanical equilibrium equation. The loss function is for volume fraction constraints. This is the physical loss weighting coefficient.

[0010] More preferably, the target loss function is set according to a preset design objective, which includes maximizing the equivalent bulk modulus, maximizing the equivalent shear modulus, minimizing the equivalent Poisson's ratio, and minimizing the average equivalent coefficient of thermal expansion. The formula for the target loss function is as follows: When the design objective is to maximize the equivalent bulk modulus, the formula for the objective loss function is:

[0011] When the design objective is to maximize the equivalent shear modulus, the formula for the objective loss function is:

[0012] When the design objective is to minimize the equivalent Poisson's ratio, the formula for the objective loss function is:

[0013] When the design objective is to minimize the average equivalent thermal expansion coefficient, the formula for the objective loss function is:

[0014] in, G is the bulk modulus, and G is the shear modulus. It is Poisson's ratio. Indicates the average equivalent thermal expansion coefficient; It represents the positive equivalent thermal expansion coefficient of the microstructure in the horizontal direction; This represents the positive equivalent thermal expansion coefficient of the microstructure in the vertical direction. It should be noted that under a two-dimensional uniform temperature field, the macroscopic equivalent deformation of the microstructure does not include shear effects; that is, the equivalent shear thermal expansion coefficient component is always 0. Therefore, the average equivalent thermal expansion coefficient is calculated only from the two positive components.

[0015] More preferably, the formula for the physical loss function is as follows:

[0016] in, This represents the total number of configuration points sampled within the microstructure computational domain; This represents the summation index of the configuration points, with a value of arrive ; Indicates the first The spatial two-dimensional coordinates of each configuration point; Indicates the first Configuration points This corresponds to the loading condition. Strong-form residuals of partial differential equations in mechanics; More preferably, the formula for calculating the strong-form residual is as follows:

[0017] in, Indicates at spatial coordinate points This corresponds to the loading condition. The mechanical equilibrium equations in strong form residuals; The divergence operator represents the differentiation operation with respect to spatial coordinates; Indicates at spatial coordinate points This corresponds to the loading condition. The stress tensor field inside the microstructure.

[0018] More preferably, the formula for the loss function corresponding to the volume fraction constraint is as follows:

[0019] in, This represents the total volume of the microstructure design domain; Represents the microstructure design domain; A collection representing phases of a solid material; An index representing the phase of a solid material; Indicates the first A type of physical material phase at spatial coordinate points The distribution weights at each location; This represents the preset target volume fraction of total physical material.

[0020] More preferably, both the material distribution network and the displacement field network adopt a multi-layer fully connected neural network structure, wherein the SIREN sinusoidal representation network is used as the backbone network and the hidden layer adopts a sinusoidal activation function.

[0021] According to another aspect of the present invention, a multi-material microstructure topology optimization system based on dual-network PINN is provided, the system including an actuator for performing the above-described multi-material microstructure topology optimization method based on dual-network PINN.

[0022] According to another aspect of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, the computer program being used to implement the above-described method for topology optimization of multi-material microstructures based on dual-network PINN.

[0023] According to another aspect of the present invention, a computer program product is provided, comprising a computer program that, when executed by a processor, implements the above-described method for topology optimization of multi-material microstructures based on dual-network PINN.

[0024] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art: 1. This invention proposes a topology optimization method based on dual-network physical information of material distribution network and displacement field network. This method uses a continuous implicit representation of multiphase material distribution and utilizes the local elastic tensor field and local thermal expansion coefficient field to obtain physical constraints and objective loss function, thereby solving the problem of unified optimization design of elastic performance and thermal expansion performance, and realizing multi-material microstructure optimization design oriented towards elastic performance and thermal expansion performance.

[0025] 2. This invention automatically calculates the derivative of the displacement field with respect to spatial coordinates, constructs a strong-form residual of the mechanical equilibrium equation, and introduces the strong-form residual as a physical constraint into the displacement field solution and optimization process. This makes the obtained displacement field, strain field and stress field more consistent with the mechanical equilibrium relationship, reduces the non-physical solutions or local imbalances that may occur when optimization is based solely on macroscopic performance targets, thereby improving the physical consistency, interpretability and reliability of the design results.

[0026] 3. This invention employs a multi-layer fully connected neural network structure to construct the material distribution network and displacement field network, reducing the reliance on point-by-point updates of explicit discrete design variables. The material distribution network takes spatial coordinates as input and continuously outputs material distribution information at corresponding locations, thus transforming the traditional unit-by-unit, point-by-point update method based on explicit discrete design variables into a holistic update method based on network parameters. Since changes in a single network parameter can simultaneously affect the material distribution at multiple spatial locations, the reliance on independent point-by-point updates of explicit discrete design variables is reduced, which is beneficial for obtaining multi-material microstructure results with better continuity and more stable interface representation.

[0027] 4. This invention employs a dual-network optimization strategy that alternately updates the material distribution network and the displacement field network, which is beneficial for improving the solution stability of topology optimization for complex multi-material microstructures; it is applicable to the design of various functional microstructures with high bulk modulus, high shear modulus, negative Poisson's ratio, and negative thermal expansion. Attached Figure Description

[0028] Figure 1 This is a flowchart of a multi-material microstructure topology optimization method based on dual-network PINN constructed according to a preferred embodiment of the present invention.

[0029] Figure 2 This is a schematic diagram of continuous interpolation and local parameter mapping of multiphase materials constructed according to a preferred embodiment of the present invention.

[0030] Figure 3 This is a flowchart of the optimization solution constructed according to a preferred embodiment of the present invention, which employs a dual-network alternating update strategy.

[0031] Figure 4 It is a dual-material microstructure for improving bulk modulus constructed according to the preferred embodiment 1 of the present invention using a topology optimization algorithm, wherein (a) is the single-cell microstructure of embodiment 1, and (b) is the periodic array structure of the single-cell microstructure of embodiment 1.

[0032] Figure 5 The bulk modulus during the optimization process constructed according to preferred embodiment 1 of the present invention is The convergence curve and schematic diagram of the microstructure topological evolution process.

[0033] Figure 6 This is a distribution diagram of displacement, strain, and stress field of a microstructure with maximized bulk modulus constructed according to preferred embodiment 1 of the present invention.

[0034] Figure 7The preferred embodiment of the present invention is a dual-material microstructure for improving shear stiffness, which is solved by topology optimization algorithm. (a) is the unit cell microstructure of Example 2, and (b) is the periodic array structure of the unit cell microstructure of Example 2.

[0035] Figure 8 The shear modulus during the optimization process constructed according to the preferred embodiment 2 of the present invention is The convergence curve and the microstructure topological evolution process diagram.

[0036] Figure 9 This is a distribution diagram of displacement, strain, and stress field of a microstructure with maximized shear modulus constructed according to preferred embodiment 2 of the present invention.

[0037] Figure 10 It is a dual-material microstructure oriented to negative Poisson's ratio response constructed according to the preferred embodiment 3 of the present invention using a topology optimization algorithm, wherein (a) is the single-cell microstructure of embodiment 3, and (b) is the periodic array structure of the single-cell microstructure of embodiment 3.

[0038] Figure 11 This is a diagram showing the convergence curve of Poisson's ratio and the evolution of microstructure topology during the optimization process, constructed according to the preferred embodiment 3 of the present invention.

[0039] Figure 12 This is a distribution diagram of displacement, strain, and stress field of a negative Poisson's ratio microstructure constructed according to preferred embodiment 3 of the present invention.

[0040] Figure 13 The present invention describes a dual-material microstructure for negative Poisson's ratio response, constructed according to a preferred embodiment of the present invention and solved using a topology optimization algorithm. (a) is a single-cell microstructure of Example 4, and (b) is a periodic array structure of the single-cell microstructure of Example 4.

[0041] Figure 14 It is a convergence curve of the thermal expansion coefficient and a diagram of the microstructure topology evolution process constructed according to the preferred embodiment 4 of the present invention.

[0042] Figure 15 This is a diagram showing the displacement, strain, and stress field distribution of a negative thermal expansion microstructure constructed according to preferred embodiment 4 of the present invention. Detailed Implementation

[0043] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0044] like Figure 1 As shown, a topology optimization method for multi-material microstructures based on a dual-network physical information neural network is characterized by the following steps: S1 Constructs a dual-network physical information neural network (1) Establish a material distribution network -Net, for design domain The input is the internal spatial coordinates (x, y), and the output is the distribution weights of each phase of material and the void phase at each location. This forms a multiphase material distribution weight field.

[0045] like Figure 2 As shown, material distribution network -Net employs a multi-layer fully connected neural network structure, with input being two-dimensional spatial coordinates. Output the distribution weights of each phase material and the void phase. ,in , This is the sum of the number of material phases and the total number of void phases. The network output layer uses the Softmax normalization function. Represented as:

[0046] in, Indicates the location of the material distribution network. The original response before Softmax normalization of the output for the m-th phase material, satisfying:

[0047] (2) Based on the multiphase material distribution weight field, determine the design domain using a multi-material continuous interpolation model. Local elastic tensor fields at various locations within the area and local thermal expansion coefficient field .

[0048] (3) Constructing a displacement field network -Net, with spatial coordinates and the weighted field of the multiphase material distribution. As input, output the microscopic displacement field at the corresponding location under different macroscopic test conditions. .

[0049] Displacement field network -Net employs a multi-layer fully connected neural network structure, using spatial coordinates and the weight of multiphase material distribution at the corresponding location As input, output the two-dimensional microscopic displacement field under the current macroscopic test conditions. Both the material distribution network and the displacement field network use the sinusoidal representation network SIREN as the backbone network, and their hidden layers use sinusoidal activation functions.

[0050] like Figure 3 As shown, in terms of training methods, the material distribution network -Net and Displacement Field Network -Net employs a phased, alternating optimization strategy. During the pre-training phase, the material distribution network... -Net is updated using the Adam optimizer to make its output approximate the preset initial material distribution; Displacement field network -Net in fixed Under the -Net condition, the Adam optimizer is used for updates to establish the initial displacement response mapping relationship under the current material distribution. In the main optimization phase, the displacement field network... -Net employs the Adam optimizer for several updates to reduce physical losses and satisfy the physical equilibrium constraints corresponding to the current material distribution; Material distribution network -Net is updated using the L-BFGS optimizer to minimize the total loss function and drive the material topology evolution.

[0051] S2 Construct the total loss function (1) Based on the automatic differentiation technique, the micro displacement field is differentiated with respect to the spatial coordinates to calculate the micro strain field. Combined with the macroscopic test strain and the local elastic tensor field, the stress field is calculated to construct the physical loss function corresponding to the strong form residual of the mechanical equilibrium equation.

[0052] The process of constructing the physical loss function includes: in the design domain Several collocation points were obtained through internal random sampling. The multiphase material distribution weights at each collocation point are obtained through the material distribution network. The position of each collocation point at the th displacement field network is obtained. Microscopic displacement field under macroscopic test conditions The displacement gradient of the microscopic displacement field with respect to spatial coordinates is calculated using automatic differentiation. And calculate the micro-strain tensor based on the small deformation geometry:

[0053] Then, macroscopic test strain The total strain tensor obtained by superposition is:

[0054] Based on the local elastic tensor field obtained from the multi-material continuous interpolation model The stress tensor is calculated as follows:

[0055] By calculating the divergence of the stress tensor, the mechanical equilibrium equations are constructed as follows:

[0056] The corresponding strong-form residual is:

[0057] The sum of squares or mean square of the strong-form residuals at all collocation points under each macroscopic test condition is used as the physical loss function. Its expression is

[0058] in, This represents the total number of points.

[0059] Macroscopic strain testing is used to apply external loading to microscopic unit cells and prevent the displacement field network from converging to trivial zero solutions. Macroscopic strain testing includes uniaxial strain testing. Directional tension condition, single axis At least one of the directional tensile and shear loads, denoted in Voigt notation as follows:

[0060] The constraint loss function is incorporated into the total loss function using the augmented Lagrangian method; The method for determining the discrete multi-material phase distribution results is as follows: at each discrete location, select:

[0061] The corresponding material phase or void phase is used as the final category for that location.

[0062] (2) Based on the multi-material continuous interpolation model and homogenization theory, calculate the equivalent elastic properties and equivalent thermal expansion properties of the microstructure, and construct the target loss function according to the preset design target.

[0063] The target loss function is constructed based on homogenization theory and specifically includes: The multiphase material distribution weight field is mapped to a local elastic tensor field through a multiphase material continuous interpolation model. and local thermal expansion coefficient field The local elastic tensor field is represented as:

[0064] The local thermal expansion coefficient field is expressed as:

[0065] in, For the first The elastic tensor of the phase material, For the first The thermal expansion coefficient of the phase material, Corresponding elastic interpolation penalty index.

[0066] The characteristic displacement field of a representative volume element is solved under periodic boundary conditions, and the equivalent elastic tensor is calculated based on homogenization theory. and equivalent thermal expansion coefficient .

[0067] in , .

[0068] A target loss function is constructed based on a preset design objective. The design objective includes at least one of maximizing the equivalent bulk modulus, maximizing the equivalent shear modulus, minimizing the equivalent Poisson's ratio, and minimizing the average equivalent coefficient of thermal expansion, wherein: the bulk modulus is... shear modulus Poisson's ratio The corresponding objective loss functions are expressed as follows:

[0069] or

[0070] or

[0071] or .

[0072] (3) The physical loss function, the objective loss function, and the constraint loss function are combined to form the total loss function, and a dual-network alternating optimization strategy is adopted for the material distribution network. -Net and displacement field network The parameters of -Net are iteratively updated to obtain an optimized multi-material distribution weight field that meets physical constraints and target performance requirements. The total loss function consists of the objective loss function, the physical loss function, and the constraint loss function, and is expressed as:

[0073] in, The target loss function is constructed based on the homogenization performance. This is the physical loss function corresponding to the residuals of the mechanical equilibrium equation. This is the constraint loss function corresponding to volume fraction constraints, performance constraints, or symmetry constraints. This is the physical loss weighting coefficient.

[0074] The constraint loss function includes at least one of volume fraction constraints, target performance constraints, and symmetry constraints, wherein the volume fraction constraint can be expressed as:

[0075] S3 selects the material phase or void phase with the largest weight at each discrete grid position as the final category at that position based on the optimized multi-material phase distribution weight field, and outputs the optimized microstructure.

[0076] The dual-network alternating optimization strategy includes: first, pre-training the material distribution network to match its output with a preset initial material distribution; then, fixing the material distribution network and pre-training the displacement field network to reduce the physical loss function; and finally, alternating between fixing one network and updating the parameters of the other during the main optimization phase. Specifically, when fixing the material distribution network, the parameters of the displacement field network are updated. Its update method is as follows:

[0077] When the displacement field network is fixed, update the material distribution network parameters. Its update method is as follows:

[0078] in, and The learning rates are set for the displacement field network and the material distribution network, respectively; the alternating update process is repeated until the convergence condition is met.

[0079] After optimization, the discrete multi-material phase distribution is determined based on the material weights at each location, and the optimized multi-material microstructure is output.

[0080] The method of the present invention will be further described below with reference to specific embodiments. Unless otherwise stated, the following embodiments all use the same material parameters, network parameters, optimization algorithm parameters, and design domain settings.

[0081] 1. Material parameter settings This embodiment involves a three-phase material system: voids, material 1, and material 2. The material properties of each phase are set as shown in Table 1 below:

[0082] 2. Network parameter settings Neural network parameters: A dual-network architecture for co-evolution is employed. Unless otherwise specified, the material distribution network... -Net and Displacement Field Network - All networks use the SIREN architecture. The material distribution network input is spatial coordinates. The output, after Softmax normalization, yields the multiphase material distribution; the input to the displacement field network is... The output is a microscopic displacement fluctuation field. Both networks have 6 hidden layers, with 80 neurons per layer. The first layer has a frequency factor. It is version 2.0. -Net uses L-BFGS for iterative updates with a learning rate of 0.01. -Net uses the Adam optimizer to update the learning rate to 0.0001.

[0083] 3. Optimize parameter settings Optimize algorithm parameters: Adaptive augmented Lagrange method , and SIMP interpolation penalty factor The initial value is 2, the maximum value is 10, and the step size is 0.5. The physical loss weight coefficient is set to... .

[0084] 4. Design Domain and Grid Settings Design Domain and Mesh: The design domain of a unit cell is taken as the side length. A square with a grid resolution of The collocation points for the physical loss function are generated within the design domain using a uniform random sampling method, with a total of 500 collocation points. Unless otherwise stated, all embodiments apply XY mirror symmetry constraints and 45° diagonal symmetry constraints.

[0085] Example 1: Microstructure Design with Maximized Bulk Modulus Using Two Materials In this embodiment, the design objective is to maximize the equivalent bulk modulus, and the volume fraction constraints for both material 1 and material 2 are set to 25%. Unless otherwise specified, the other material parameters, network parameters, optimization parameters, and design domain settings are the same as the aforementioned unified parameter settings. The solution is obtained using the dual-network alternating optimization method, resulting in a dual-material microstructure aimed at improving the bulk modulus. The optimization results are as follows: Figure 4 As shown, by Figure 4 As shown in (a), the final optimized unit cell microstructure forms a load-bearing skeleton with a central cross connection and diagonal support as the main components. The high modulus material is mainly distributed in the central connection area and the diagonal main load-bearing path, while the low modulus material is mainly distributed in the outer periphery of the skeleton and near the interface between the two phases. Figure 4 Image (b) shows the array structure of the unit cell after periodic tiling, indicating that the obtained microstructure can form a continuous and stable periodic topological configuration. Figure 5As shown, in the initial stage of optimization, the bulk modulus initially decreases, then increases continuously as the material is redistributed and the main load-bearing skeleton gradually takes shape, eventually stabilizing. Simultaneously, the microstructure topology gradually evolves from the initial configuration into a more clearly defined dual-material distribution structure, indicating that the proposed dual-network alternating optimization strategy can effectively drive the material distribution towards a high bulk modulus target. Figure 6 As shown, the displacement, strain, and stress fields are mainly distributed along the central connection, diagonal support ribs, and material interface regions. The stress concentration areas correspond to the main load-bearing paths, indicating that this configuration can improve the overall resistance to volumetric deformation by using high-modulus materials to bear the main load and low-modulus materials to participate in local coordination and interface transition. This demonstrates that the method of this invention can obtain bimaterial microstructures oriented towards improving bulk modulus.

[0086] Example 2: Microstructure Design with Maximized Shear Modulus in Two Materials In this embodiment, the design objective is to maximize the equivalent shear modulus. The volume fraction constraints for both material 1 and material 2 are set to 25%, and the remaining parameters are the same as in Embodiment 1. After solving using the described dual-network alternating optimization method, a bimaterial microstructure oriented towards improving shear stiffness is obtained. Figure 7 As shown in (a), the final optimized unit cell microstructure forms an X-shaped main load-bearing skeleton that unfolds along two diagonals. The high modulus material is mainly distributed in the diagonal support main path and the central connection area, while the low modulus material is mainly distributed in the outer covering area and the interface transition area. Figure 7 Image (b) shows the array structure of the unit cell after periodic tiling, indicating that the obtained microstructure can form a continuous periodic diagonal support network. Figure 8 As shown, in the initial stage of optimization, the equivalent shear modulus initially decreases slightly, then increases continuously as the diagonal main load-bearing skeleton becomes clearer and the material is redistributed, eventually stabilizing. Simultaneously, the microstructure topology gradually evolves from the initial distribution to a more clearly defined X-shaped dual-material configuration, indicating that the proposed dual-network alternating optimization strategy can effectively drive the material distribution towards a high shear modulus target. Figure 9 As shown, the displacement, strain, and stress fields are mainly concentrated in the X-shaped diagonal support ribs, the central connection, and the material interface region. The stress concentration region coincides with the main shear force transmission path, indicating that this configuration can effectively transfer macroscopic shear deformation to the diagonal load-bearing skeleton. Furthermore, it improves the overall shear resistance by using a high-modulus material to bear the main shear load and a low-modulus material to participate in local coordination and interface transition. This demonstrates that the method of this invention can obtain a bimaterial microstructure oriented towards improving shear stiffness.

[0087] Example 3: Design of a dual-material negative Poisson's ratio microstructure In this embodiment, the design objective is to minimize the equivalent Poisson's ratio. The volume fraction constraints for both material 1 and material 2 are set to 25%, and the remaining parameters are the same as in Example 1. After solving using the described dual-network alternating optimization method, a dual-material microstructure with a negative Poisson's ratio response is obtained. Figure 10 As shown in (a), the final optimized unit cell microstructure forms a framework configuration with obvious concave features. The central region is a relatively stable internal unit, and the external skeleton is connected to the center by thin connecting ribs. The high modulus material and the low modulus material together form a rotatable and openable composite skeleton along the concave boundary and the connecting rib region. Figure 10 Image (b) shows the array structure of the unit cell after periodic tiling, indicating that the obtained microstructure can form a continuous periodic negative Poisson's ratio configuration. Figure 11 As shown, in the initial stage of optimization, the equivalent Poisson's ratio rapidly decreases from near zero to a negative value, and then further decreases as the concave skeleton and connecting rib structure gradually become clearer, eventually stabilizing. This indicates that the described dual-network alternating optimization strategy can effectively drive the material distribution towards the negative Poisson's ratio target. Figure 12 As shown, the displacement, strain, and stress fields are mainly concentrated in the concave connecting ribs, the central connecting region, and the outer turning points. The high-stress regions correspond to the main deformation regions, indicating that this configuration, under load, can achieve a lateral expansion response through the opening of the concave units, the coordinated rotation of the connecting rods, and local bending deformation, thereby generating a negative Poisson's ratio effect. This demonstrates that the method of this invention can obtain a bimaterial microstructure with a negative Poisson's ratio response.

[0088] Example 4: Negative Thermal Expansion Microstructure Design In this embodiment, the design objective is to minimize the average equivalent coefficient of thermal expansion. The volume fraction constraint for both material 1 and material 2 is set to 25%, and the coefficient of thermal expansion parameters in Table 1 are enabled. The remaining parameter settings are the same as in Example 1. After solving using the described dual-network alternating optimization method, a dual-material microstructure with a macroscopic negative thermal expansion response is obtained. Figure 13 As shown in (a), the final optimized unit cell microstructure forms a composite skeleton structure with obvious inward features. The central region is connected to the external material skeleton through oblique connecting ribs. High thermal expansion material and low thermal expansion material are alternately distributed along the connecting ribs and interface region, thus forming a dual-material thermal response unit that can undergo synergistic deformation under heating conditions. Figure 13 Image (b) shows the array structure of the unit cell after periodic tiling, indicating that the obtained microstructure can form a continuous periodic negative thermal expansion configuration. Figure 14As shown, in the initial stage of optimization, the average equivalent thermal expansion coefficient fluctuates to some extent, then rapidly decreases to a negative value as the dual-material thermal response framework gradually forms, and tends to stabilize in subsequent iterations. The final thermal expansion coefficient converges to -3.656, indicating that the proposed dual-network alternating optimization strategy can effectively drive the material distribution towards the negative thermal expansion target. Figure 15 As shown, the displacement, strain, and stress fields are mainly concentrated in the central connecting region, the oblique connecting ribs, and the material interface region. The high-response region corresponds to the main deformation path induced by thermal mismatch, indicating that this configuration, under temperature load, can offset the local positive thermal expansion effect through local bending, rotation, and inward deformation caused by the difference in thermal expansion between different materials, thereby achieving a macroscopic negative thermal expansion response. This demonstrates that the method of this invention can obtain a bimaterial microstructure with a macroscopic negative thermal expansion response.

[0089] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A topology optimization method for multi-material microstructures based on dual-network PINN, characterized in that, The method includes the following steps: A material distribution network is established, with the coordinates of each point in the design domain of the microstructure to be optimized as the output and the multiphase material distribution weights of the microstructure to be optimized as the input; at the same time, a displacement field network is established, with the multiphase material distribution weights of the microstructure to be optimized as the input and the microscopic displacement fields at various points on the microstructure to be optimized as the output. The local elastic tensor field and local thermal expansion coefficient field within the design domain are calculated using the obtained multiphase material distribution weights and micro displacement fields. The strong form residuals at various points within the design domain are calculated using the local elastic tensor field and local thermal expansion coefficient field, and the physical loss function of the design domain is constructed using the strong form residuals. At the same time, the target loss function is constructed using the local elastic tensor field and local thermal expansion coefficient field. A loss function with respect to volume fraction constraints, a target loss function, and a total loss function with respect to physical loss functions are established. The network parameters of the material distribution network and the displacement field network are adjusted to minimize the total loss function, thereby obtaining the multiphase material distribution weights at the minimum loss function, thus achieving topology optimization of the microstructure.

2. The method for topology optimization of multi-material microstructures based on dual-network PINN as described in claim 1, characterized in that, The formula for the total loss function is as follows: in, The target loss function is constructed based on the homogenization performance. This is the physical loss function corresponding to the residuals of the mechanical equilibrium equation. The loss function is for volume fraction constraints. This is the physical loss weighting coefficient.

3. The method for topology optimization of multi-material microstructures based on dual-network PINN as described in claim 2, characterized in that, The target loss function is set according to a preset design objective, which includes maximizing the equivalent bulk modulus, maximizing the equivalent shear modulus, minimizing the equivalent Poisson's ratio, and minimizing the average equivalent coefficient of thermal expansion. The formula for the target loss function is as follows: When the design objective is to maximize the equivalent bulk modulus, the formula for the objective loss function is: When the design objective is to maximize the equivalent shear modulus, the formula for the objective loss function is: When the design objective is to minimize the equivalent Poisson's ratio, the formula for the objective loss function is: When the design objective is to minimize the average equivalent thermal expansion coefficient, the formula for the objective loss function is: ， in, G is the bulk modulus, and G is the shear modulus. It is Poisson's ratio. This represents the average equivalent thermal expansion coefficient. This represents the positive equivalent thermal expansion coefficient of the microstructure in the horizontal direction. It represents the positive equivalent thermal expansion coefficient of the microstructure in the vertical direction.

4. The multi-material microstructure topology optimization method based on dual-network PINN as described in claim 3, characterized in that, The formula for the physical loss function is as follows: in, This represents the total number of placement points sampled within the microstructure computational domain. This represents the summation index of the configuration points, with a value of arrive ; Indicates the first The spatial two-dimensional coordinates of each configuration point Indicates the first Configuration points This corresponds to the loading condition. The strong-form residuals of the partial differential equations of mechanics.

5. The multi-material microstructure topology optimization method based on dual-network PINN as described in claim 4, characterized in that, The formula for calculating the strong-form residual is as follows: in, Indicates at spatial coordinate points This corresponds to the loading condition. The mechanical equilibrium equations have strong form residuals. The divergence operator represents the differentiation operation with respect to spatial coordinates. Indicates at spatial coordinate points This corresponds to the loading condition. The stress tensor field inside the microstructure.

6. The multi-material microstructure topology optimization method based on dual-network PINN as described in claim 5, characterized in that, The formula for the loss function corresponding to the volume fraction constraint is as follows: in, Represents the total volume of the microstructure design domain; Represents the microstructure design domain. Represents a collection of phases of a solid material. Indexes representing phases of a solid material. Indicates the first A type of physical material phase at spatial coordinate points The distribution weights at that location, This represents the preset target volume fraction of total physical material.

7. A method for topology optimization of multi-material microstructures based on dual-network PINN as described in claim 1 or 6, characterized in that, Both the material distribution network and the displacement field network adopt a multi-layer fully connected neural network structure, in which the SIREN sinusoidal representation network is used as the backbone network and the hidden layer uses a sinusoidal activation function.

8. A multi-material microstructure topology optimization system based on dual-network PINN, characterized in that, The system includes an actuator for performing a multi-material microstructure topology optimization method based on dual-network PINN as described in any one of claims 1-7.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, The computer program is used to implement the multi-material microstructure topology optimization method based on dual-network PINN as described in any one of claims 1-7.

10. A computer program product, characterized in that, The method includes a computer program that, when executed by a processor, implements a multi-material microstructure topology optimization method based on dual-network PINN as described in any one of claims 1-7.

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