Anisotropic multi-material structure topological optimization method based on meshless EFGM and FNN

Abstract

The invention discloses an anisotropic multi-material structure topological optimization method based on meshless EFGM and FNN, and the method comprises the steps: (1) inputting the Poisson's ratio, Young's modulus, direction angle and other material attributes of an anisotropic material, and carrying out the discretization of a design domain through a meshless EFGM node; (2) determining network parameters such as the network depth, the number of neurons in each layer and the learning rate, and building a full-connection network model; (3) taking the coordinates of the meshless nodes as network input, and predicting the density of the meshless nodes; (4) establishing a meshless EFGM discrete control equation of the structural statics problem; (5) establishing a mathematical model for topological optimization of the anisotropic multi-material structure based on the meshless EFGM and the FNN; (6) defining a loss function according to the flexibility and volume of the current topological structure; (7) executing network training, and updating network parameters by using back propagation; and (8) inputting a termination condition, and judging whether the algorithm converges. The meshless EFGM and FNN-based topological optimization method for the anisotropic multi-material structure is high in calculation efficiency, clear in boundary of the topological structure, simple and practical.

Description

Technical Field

[0001] The present invention belongs to the field of structural optimization design in computer-aided engineering, and specifically relates to an anisotropic multi-material structure topology optimization method based on an element-free Galerkin method (EFGM) and a fully connected feedforward neural network (FNN). Background Art

[0002] In engineering structural design, traditional single-material structures are no longer able to meet the comprehensive performance requirements under complex loads and multi-physics coupling. Multi-material structural design integrates materials with varying mechanical properties in different regions, achieving complementary performance and functional integration in terms of stiffness, strength, mass, thermal conductivity, and vibration suppression. Compared to topological optimization of structures based on a single material, multi-material topology optimization allows for flexible material configuration to target specific load conditions and functional requirements, thereby improving strength, reducing weight, and enhancing structural performance, such as fatigue resistance. Topological optimization of multi-material structures has been widely applied in fields such as aerospace, automotive, and consumer electronics, driving the realization of lighter, stronger, and more functional designs. Furthermore, the advantage of anisotropic materials in topology optimization lies primarily in their ability to precisely design based on mechanical properties in different directions, thereby achieving superior structural performance. Topological optimization can fully exploit the properties of anisotropic materials to optimize the shape and material distribution of a structure, increasing strength and stiffness while reducing weight. In the engineering field, anisotropic structural topology optimization provides more flexible and efficient solutions for engineering design. For example, in aerospace, the use of anisotropic materials can design lighter and stiffer wing structures, improving fuel efficiency and flight performance. Therefore, anisotropic multi-material structural design can fully utilize the properties of anisotropic materials, realize the synergistic effect of different anisotropic materials, and provide an effective solution for high-performance, lightweight structural design.

[0003] The numerical solution methods commonly used in topology optimization include the Finite Element Method (FEM), the Finite Volume Method (FVM), and the Finite Difference Method (FDM). Although the above methods have good numerical accuracy, due to the existence of the grid, the numerical calculation results are easily affected by the deformed grid, resulting in numerical instability, and the topology optimization results may have poor convergence, checkerboard, intermediate density and other phenomena. The meshless method does not require mesh division. By discretizing a series of nodes in the computational domain and constructing shape functions using node information, it can effectively overcome mesh dependence. As a relatively mature meshless method, the Element-free Galerkin Method (EFGM) has high-order continuous shape functions. In the sensitivity analysis of topology optimization, it can rely on the original sensitivity of the objective function and obtain a clearer topological boundary without using a filter. The topology optimization of anisotropic multi-material structures based on meshless EFGM can fully utilize the mechanical properties of anisotropic materials, and meshless EFGM can achieve higher calculation accuracy, providing new solutions for the precise design of various structures.

[0004] Topology optimization methods based on meshless EFGM iterative solvers require the calculation of intermediate solutions at each iteration. For large-scale structural topology optimization problems, the time required for numerical calculations can increase exponentially, increasing computational costs. With the continuous advancement of artificial intelligence (AI) technology, it is becoming increasingly possible to combine neural networks with meshless EFGM-based topology optimization to achieve efficient and accurate topology design. A fully connected feedforward neural network is a fundamental neural network architecture, where each neuron in each layer is connected to every neuron in the next layer, and the network parameters are automatically adjusted through backpropagation to achieve a solution. Compared to data-driven neural network methods, topology optimization methods based on FNN and meshless EFGM require only the node coordinates of the EFGM as neural network input, avoiding the need for extensive upfront dataset preparation and exhibiting good generalization. Combining FNN and meshless EFGM for topology optimization of anisotropic multi-material structures can address the inefficiency of iterative solvers in meshless EFGM-based topology optimization. Furthermore, the neural network can update sensitivity through backpropagation, eliminating the need for manual sensitivity analysis. Summary of the Invention

[0005] As a lightweight structural design method, topology optimization has been widely used in the field of engineering design. As engineering structures gradually become larger and their service environments become more complex, the use of multiple materials in topological structure design has gradually become a trend, and large-scale topology optimization urgently needs an efficient and high-precision calculation method. In order to solve the problems faced by the meshless EFGM iterative solution method, such as low computational efficiency and the easy generation of intermediate density in the mesh-based topology optimization, the present invention proposes a topology optimization method for anisotropic multi-material structures based on meshless EFGM and FNN. The method utilizes meshless EFGM nodes to discretize the design domain, adopts FNN to realize the relative density field prediction of the meshless EFGM nodes in the topological structure, constructs a neural network loss function based on structural flexibility and volume constraints, and establishes a mathematical model for the topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN; and writes a computer program of the algorithm to perform topological optimization design on multi-material structures with different neural network parameters and anisotropic material property parameters, and outputs its optimal topological structure.

[0006] The present invention solves its technical problems by employing a topology optimization method for anisotropic multi-material structures based on a meshless EFGM and FNN. The method utilizes the meshless EFGM to discretize the static governing equations of the anisotropic structure. The meshless EFGM node coordinates serve as the FNN input, and the node relative density field serves as the neural network output. The network loss function is composed of two items: flexibility and volume constraints. The FNN network minimizes the loss function during training to achieve minimum flexibility and volume constraints. The topology optimization method for anisotropic multi-material structures based on the meshless EFGM and FNN automatically updates neural network parameters through backpropagation to achieve topological structure prediction. The mechanical properties of the anisotropic structure are controlled by adjusting the Poisson's ratio and material orientation angles, and the network's fitting capability is improved by adjusting the number of neural network layers and the number of neurons in each layer.

[0007] The specific implementation steps of the technical solution of the present invention are as follows: (1) According to the design requirements of the actual engineering structure, determine the initial design domain of the structure and the volume fraction η of each anisotropic material in the multi-material structure. j , enter the Young's modulus of each anisotropic material Poisson's ratio Shear modulus Poisson's ratio factor Bt j , anisotropic material direction angle θ j , given the design domain boundary conditions and load size, the calculation point information in the design domain is obtained according to the meshless node information of the design domain, the background grid and the given boundary conditions; (2) Calculate the structural stiffness matrix without introducing node relative density penalty based on meshless EFGM and apply boundary conditions: (a) Solve the elastic matrices of various materials according to the mechanical properties of anisotropic materials Where T j is the transformation matrix of the anisotropic multi-material between the meshless EFGM and FNN, where θ j is the orientation angle of the jth anisotropic material based on the meshless EFGM and FNN, as well as They are the Young's modulus and Poisson's ratio of the jth anisotropic material in the ξ and η directions in the material coordinate system, and satisfy the relationship Bt j is the Poisson's ratio factor of the jth anisotropic material, is the shear elastic modulus of the jth anisotropic material; (b) The shape function φ of the meshless EFGM is constructed using the moving least squares approximation i , and integrate at the calculation point to calculate the meshless EFGM anisotropic multi-material structure stiffness matrix without introducing node relative density penalty Where the strain matrix Elastic matrix of the jth anisotropic material Where, (c) Determine the overall force load vector F of the meshless EFGM in the design domain according to the magnitude of the force load applied in the design domain. Where the shape function matrix N(x) is the vector of MLS shape functions corresponding to the grid-free nodes in the neighborhood of the calculation point x. is the force matrix on the given boundary Γ, is the force on the design domain Ω; (d) According to the given force and displacement boundary conditions, the penalty function method is used to process various force and displacement boundary conditions to obtain the penalty term K of the meshless EFGM overall force stiffness matrix fα and the EFGM overall force load vector penalty term F α ,in, Where α is the penalty factor of the penalty function method, and the displacement constraint judgment matrix When the displacement constraint is applied in the x or y direction, the corresponding s1 and s2 are 1, otherwise they are 0. is the displacement constraint matrix; (3) Predict the relative density field of the gridless EFGM nodes of the topological structure and calculate the displacement of the topological structure: (a) Determine the number of neural network layers, the number of neurons in each layer, the activation function type, the network learning rate, build a fully connected feedforward neural network, initialize the weights and biases of the neural network, and the neural network parameters It is composed of the weight matrix w and bias matrix b of all layers Where w=[w1,w2,...,w n ],b=[b1,b2,...,b n ],w n represents the weight associated with the nth layer of neurons, b n represents the bias associated with the nth layer of neurons; (b) The meshless EFGM node coordinates are used as the input of the fully connected feedforward neural network to predict the meshless EFGM node relative density field of the anisotropic multi-material structure based on the meshless EFGM and FNN. In the formula is the matrix composed of the density of each node s represents the number of meshless EFGM nodes in the design domain, m represents the number of anisotropic material types in the design domain, x represents the meshless EFGM node coordinate matrix, BN represents batch normalization of the input information, σ represents the ReLU activation operation on the input data, and Softmax represents the Softmax activation operation on the input data; (c) The relative density of the meshless EFGM of the multi-material structure at the calculation point is obtained by interpolation based on the predicted relative density of the meshless EFGM nodes. Where, represents the relative density of the jth material at the lth meshless EFGM calculation point, N represents the number of meshless EFGM nodes in the influence domain of the lth calculation point, represents the relative density of the jth material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN; (d) the meshless EFGM overall stiffness matrix of the design domain assembled according to the relative density of the meshless EFGM calculation points ng represents the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on the meshless EFGM and FNN, and p is the penalty factor of the topology optimization SIMP method; (e) The static discrete governing equations of the meshless EFGM are established to solve the displacement parameter values ​​of the meshless EFGM nodes in the design domain. The displacement value U of each meshless EFGM node is solved according to the displacement parameter value of the meshless EFGM node in the design domain; (4) Calculate the flexibility of anisotropic multi-material structures based on meshless EFGM and FNN and update the network parameters of FNN by back propagation: (a) Calculate the flexibility of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, u l is the displacement matrix at the lth meshless EFGM calculation point, kl is the stiffness matrix at the lth meshless EFGM calculation point, is the flexibility of the jth anisotropic material at the lth meshless EFGM calculation point, and ng represents the number of meshless EFGM calculation points in the design domain; (b) Calculation of the volume of anisotropic multi-material topology based on meshless EFGM and FNN Defining the loss function of a fully connected feedforward neural network (FNN) for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the topological flexibility of the current training step in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, C0 represents the topological flexibility obtained by the first execution of training in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, V0 represents the total volume of the design domain, and v j represents the volume fraction of the jth anisotropic material, and λ is the relaxation factor of the volume constraint in the topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN; (c) FNN network training is performed, and the derivatives of the loss function with respect to the neural network parameters are calculated using back propagation. And update the network parameters, (5) The governing equations for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN are established as follows: Where w k is the neural network weight associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, b k is the neural network bias associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, n represents the number of hidden layers in the fully connected feedforward neural network FNN, The global stiffness moment of the meshless EFGM with the relative node density penalty is introduced. is the gridless displacement parameter column vector, F is the gridless EFGM overall force load column vector, K fα is the penalty term of the EFGM global stiffness matrix, F α is the penalty term of the meshless EFGM global force load column vector, φ i is the shape function of the meshless EFGM, is the relative density of the j-th material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN, is the relative density of the jth material at the lth meshless EFGM calculation point, ng is the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on meshless EFGM and FNN, N is the number of nodes in the influence domain of the lth calculation point, and η j is the volume fraction of the jth anisotropic material, V0 is the total volume of the design domain, and s represents the number of meshless EFGM nodes in the design domain; (6) Input the iteration termination condition. If the termination condition is met, the iteration is terminated, and the optimal topology structure of the anisotropic multi-material structure topology optimization based on the meshless EFGM and FNN is output according to the relative density value of each meshless node. If it is not met, continue to execute the subsequent steps and loop steps (3)-(4) until the termination condition is met.

[0008] The beneficial effects of the present invention are as follows: the present invention realizes the topological optimization design of anisotropic multi-material structures based on the meshfree Galerkin method and the fully connected feedforward neural network, which can effectively improve the high computational cost problem of the iterative meshfree EFGM topology optimization method; the present invention is based on the meshfree Galerkin method, and constructs shape functions through meshfree nodes in the local support domain, which can effectively avoid the disadvantages of the mesh-based solution method being affected by the deformed mesh, and has better numerical stability; the present invention adopts multiple anisotropic materials to realize topological structure design, which can provide more design freedom, and through the coordinated use of multiple materials, it can break through the limitations of the physical properties of a single material and design a structural form with more excellent performance; the topological structure obtained by the present invention has a clear and smooth contour, which is convenient for performance analysis and subsequent processing and manufacturing; the present invention controls the mechanical properties of the anisotropic structure through the Poisson's ratio factor and the material orientation angle, and can perform topological optimization design on the anisotropic structure according to engineering requirements, and has good design flexibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0010] Figure 1 The global coordinate system and material coordinate system of the anisotropic structure of the present invention are Figure 2 This is the design flow chart of the anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN of the present invention Figure 3 Schematic diagram of the structural design area size, load and boundary conditions of an embodiment of the present invention Figure 4The optimal topological structure obtained by using two materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt5=1, and the material orientation angle θ1=θ5=0° in the embodiment of the present invention is Figure 5 The optimal topological structure obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 6 The optimal topological structure obtained by using four materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material orientation angle θ1=θ2=θ3=θ5=0° in the embodiment of the present invention is Figure 7 The optimal topological structure obtained by using four materials (including the empty phase material) is shown in the embodiment of the present invention when the number of neural network layers h=4, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material orientation angle θ1=θ2=θ3=θ5=0°. Figure 8 The optimal topological structure obtained by using four materials (including the empty phase material) when the number of neural network layers h=6, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material orientation angle θ1=θ2=θ3=θ5=0° is shown in FIG. Figure 9 The optimal topological structure obtained by using four materials (including the empty phase material) when the number of neural network layers h=10, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material orientation angle θ1=θ2=θ3=θ5=0° is shown in FIG. Figure 10 The optimal topological structure obtained by using four materials (including the empty phase material) is shown in the embodiment of the present invention when the number of neural network layers h=12, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material orientation angle θ1=θ2=θ3=θ5=0°. Figure 11The optimal topological structure obtained by using four materials (including the empty phase material) is shown in the embodiment of the present invention when the number of neural network layers h=14, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material orientation angle θ1=θ2=θ3=θ5=0°. Figure 12 The optimal topological structure obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.003, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 13 The optimal topological structure obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.005, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 14 The optimal topological structure obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.007, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 15 The optimal topological structure obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.009, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 16 The optimal topological structure obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.011, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 17 The optimal topological structure obtained by using three materials (including the empty phase material) is shown in the embodiment of the present invention, wherein the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, and the material orientation angles θ1=θ4=θ5=0°. Figure 18The optimal topological structure obtained by using three materials (including the void phase material) is shown in the embodiment of the present invention when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=30°. Figure 19 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=60° in the embodiment of the present invention is Figure 20 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, and the material orientation angles θ1=θ5=0° and θ4=90° is shown in FIG. Figure 21 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=120° in the embodiment of the present invention is Figure 22 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, and the material orientation angles θ1=θ5=0° and θ4=150° is shown in FIG. Figure 23 The optimal topological structure 3D printing model obtained by using two materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt5=1, and the material direction angle θ1=θ5=0° in the embodiment of the present invention is Figure 24 The optimal topological structure 3D printing model obtained by using three materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson ratio factor Bt1=Bt2=Bt5=1, and the material direction angle θ1=θ2=θ5=0° in the embodiment of the present invention is Figure 25This is an optimal topological structure 3D printing model obtained by using four materials (including void phase material) when the number of neural network layers h = 8, the number of neurons in each layer n = 80, the learning rate lr = 0.001, the Poisson's ratio factor Bt1 = Bt2 = Bt3 = Bt5 = 1, and the material orientation angle θ1 = θ2 = θ3 = θ5 = 0° in an embodiment of the present invention. DETAILED DESCRIPTION

[0011] See also Figure 1 and Figure 2 The anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN mainly includes the following steps: First, determine the Young's modulus of each anisotropic material Poisson's ratio Shear modulus Poisson's ratio factor Bt j , anisotropic material direction angle θ j According to Hooke's law, when there is a material orientation angle θ between the global coordinate system xy and the material coordinate system ξ-η j When , the relationship between stress and strain of the jth anisotropic material is Where, The elastic matrix of the jth anisotropic material, T j for The coordinate transformation matrix, ,in, as well as They are the Young's modulus and Poisson's ratio of the jth anisotropic material in the ξ and η directions in the material coordinate system, and satisfy the relationship is the shear elastic modulus of the jth anisotropic material. The Poisson's ratio factor is defined as Then when Bt j =1,θ j = 0 is an isotropic material.

[0012] The shape function φ of the meshless EFGM is constructed using the Moving Least Square Approximation (MLS). i , and integrate at the calculation point to calculate the meshless EFGM anisotropic multi-material structure stiffness matrix without introducing node relative density penalty Where the strain matrix Elastic matrix of the jth anisotropic material Where, The meshless EFGM overall force load vector F of the design domain is determined according to the magnitude of the force load applied in the design domain. Where the shape function matrix N(x) is the vector of MLS shape functions corresponding to the grid-free nodes in the neighborhood of the calculation point x. is the force matrix on the given boundary Γ, is the force on the design domain Ω; according to the given force and displacement boundary conditions, the penalty function method is used to process various force and displacement boundary conditions to obtain the penalty term K of the meshless EFGM overall force stiffness matrix fα and the EFGM overall force load vector penalty term F α ,in, Where α is the penalty factor of the penalty function method, and the displacement constraint judgment matrix When the displacement constraint is applied in the x or y direction, the corresponding s1 and s2 are 1, otherwise they are 0. is the displacement constraint matrix.

[0013] Predict the relative density field of the meshless EFGM nodes of the topological structure and calculate the displacement of the topological structure: (a) Determine the number of neural network layers, the number of neurons in each layer, the activation function type, the network learning rate, build a fully connected feedforward neural network, initialize the weights and biases of the neural network, and the neural network parameters It is composed of the weight matrix w and bias matrix b of all layers Where w=[w1,w2,...,w n ],b=[b1,b2,...,b n ],w n represents the weight associated with the nth layer of neurons, b n represents the bias associated with the nth layer of neurons; (b) The meshless EFGM node coordinates are used as the input of the fully connected feedforward neural network to predict the meshless EFGM node relative density field of the anisotropic multi-material structure based on the meshless EFGM and FNN. In the formula is the matrix composed of the density of each node s represents the number of meshless EFGM nodes in the design domain, m represents the number of anisotropic material types in the design domain, x represents the meshless EFGM node coordinate matrix, BN represents batch normalization of the input information, σ represents the ReLU activation operation on the input data, and Softmax represents the Softmax activation operation on the input data; (c) The relative density of the meshless EFGM of the multi-material structure at the calculation point is obtained by interpolation based on the predicted relative density of the meshless EFGM nodes. Where, represents the relative density of the jth material at the lth meshless EFGM calculation point, N represents the number of meshless EFGM nodes in the influence domain of the lth calculation point, represents the relative density of the jth material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN; (d) the meshless EFGM overall stiffness matrix of the design domain assembled according to the relative density of the meshless EFGM calculation points ng represents the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on the meshless EFGM and FNN, and p is the penalty factor of the topology optimization SIMP method; (e) The static discrete governing equations of the meshless EFGM are established to solve the displacement parameter values ​​of the meshless EFGM nodes in the design domain. The displacement value U of each meshless EFGM node is solved based on the displacement parameter value of the meshless EFGM node in the design domain.

[0014] Computing the compliance of anisotropic multi-material structures based on meshless EFGM and FNN and updating the network parameters of FNN by backpropagation: (a) Computing the compliance of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, u l is the displacement matrix at the lth meshless EFGM calculation point, k l is the stiffness matrix at the lth meshless EFGM calculation point, is the flexibility of the jth anisotropic material at the lth meshless EFGM calculation point, and ng represents the number of meshless EFGM calculation points in the design domain; (b) Calculation of the volume of anisotropic multi-material topology based on meshless EFGM and FNN Defining the loss function of a fully connected feedforward neural network (FNN) for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the topological flexibility of the current training step in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, C0 represents the topological flexibility obtained by the first execution of training in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, V0 represents the total volume of the design domain, and v j represents the volume fraction of the jth anisotropic material, and λ is the relaxation factor of the volume constraint in the topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN; (c) FNN network training is performed, and the derivatives of the loss function with respect to the neural network parameters are calculated using back propagation. And update the network parameters,

[0015] The governing equations for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN are established as follows: Where w k is the neural network weight associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, b k is the neural network bias associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, n represents the number of hidden layers in the fully connected feedforward neural network FNN, The global stiffness moment of the meshless EFGM with the relative node density penalty is introduced. is the gridless displacement parameter column vector, F is the gridless EFGM overall force load column vector, K fα is the penalty term of the EFGM global stiffness matrix, F α is the penalty term of the meshless EFGM global force load column vector, φ i is the shape function of the meshless EFGM, is the relative density of the j-th material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN, is the relative density of the jth material at the lth meshless EFGM calculation point, ng is the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on meshless EFGM and FNN, N is the number of nodes in the influence domain of the lth calculation point, and η j is the volume fraction of the jth anisotropic material, V0 is the total volume of the design domain, and s represents the number of meshless EFGM nodes in the design domain.

[0016] Input the iteration termination condition. If the termination condition is met, the iteration terminates and the optimal topology of the anisotropic multi-material structure based on the meshless EFGM and FNN is output according to the relative density value of each meshless node. If it is not met, loop steps

[0013] and

[0014] until the termination condition is met.

[0017] See also Figure 2 The specific steps of the anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN are as follows: (1) According to the design requirements of the actual engineering structure, determine the initial design domain of the structure, the volume fraction η of each anisotropic material j , enter the Young's modulus of the anisotropic material Poisson's ratio Shear modulus Poisson's ratio factor Bt j , anisotropic material direction angle θ j , calculate the elastic matrix of anisotropic materials (2) According to the theory of the element-free Galerkin method, the moving least squares method is used to construct the shape function to approximate the unknown field function. The MLS approximate expression of u(x) at x is defined as Where a Τ (x)=[a1(x),a2(x),…,a m (x)],a i (x) is the basis function, m is the number of terms in the basis function, and a is selected as the linear basis. Τ (x) = [1, x, y], λ(x) is a vector of unknown coefficients consisting of a set of functions of x; (3) The unknown coefficient vector λ(x) is obtained by minimizing the functional J, which is Where u i is the function value of the function u(x) at the gridless node in the neighborhood of the calculation point x, NP is the number of gridless nodes in the neighborhood of the calculation point x, ω(xx i ) is the weight function, and the cubic spline weight function is selected. Its specific expression is Where r = || xx i || / d mI , calculate the neighborhood radius of point x as d mI =s×d k , s is a multiplier greater than 1, d k is the distance between gridless node i and the kth gridless node closest to it; (4) Therefore, λ(x) is λ(x)=A -1 (x)B(x)u (8) Where, B(x)=[ω1(x)a(x1)ω2(x)a(x2)···ω NP (x)a(x NP )],u=[u1u2 u3…u NP ] T ; (5) Substituting formula (8) into formula (5), we can get, Where N(x) is the vector of MLS shape functions corresponding to the grid-free nodes in the neighborhood of the calculation point x, N(x) = [N1(x)N2(x)···N NP (x)]=a T (x)A -1 (x)B(x); (6) According to the defined mesh-free support domain radius, find the nodes within the influence domain of the calculation point; (7) Calculation of the stiffness matrix of the meshless EFGM anisotropic multi-material structure without introducing node relative density penalty Where the strain matrix Elastic matrix of the jth anisotropic material In the formula Given the design domain boundary conditions and load size, the penalty function method is used to process various force and displacement boundary conditions to obtain the stiffness matrix penalty term K fα and force load penalty term F α ; In the formula, the shape function matrix α is the penalty factor of the penalty function method, and the displacement constraint judgment matrix When the displacement constraint is applied in the x or y direction, the corresponding s1 and s2 are 1, otherwise they are 0; (8) Determine the number of neural network layers, the number of neurons in each layer, the activation function type, the network learning rate, build a fully connected feedforward neural network, initialize the weights and biases of the neural network, and the neural network parameters It is composed of the weight matrix w and bias matrix b of all layers Where w=[w1,w2,...,w n ],b=[b1,b2,...,b n ],w n represents the weight associated with the nth layer of neurons, b n represents the bias associated with the nth layer of neurons; (9) The meshless EFGM node coordinates are used as the input of the fully connected feedforward neural network to predict the meshless EFGM node relative density field of the anisotropic multi-material structure based on the meshless EFGM and FNN. In the formula is the matrix composed of the density of each node s represents the number of meshless EFGM nodes in the design domain, m represents the number of anisotropic material types in the design domain, x represents the meshless EFGM node coordinate matrix, BN represents batch normalization of the input information, σ represents the ReLU activation operation performed on the input data, and Softmax represents the Softmax activation operation performed on the input data; (10) The relative density of the multi-material structure meshless EFGM at the calculation point is obtained by interpolating the predicted relative density of the multi-material structure meshless EFGM node. Where, represents the relative density of the jth material at the lth meshless EFGM calculation point, N represents the number of meshless EFGM nodes in the influence domain of the lth calculation point, Represents the relative density of the jth material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN; assembles the meshless EFGM global stiffness matrix of the design domain according to the relative density of the meshless EFGM calculation points, and establishes the meshless EFGM discrete control equation for the structural statics problem Where, To introduce the relative density of the meshless EFGM global force stiffness matrix, K fα is the penalty term of the meshless EFGM global force stiffness matrix, is the gridless displacement parameter column vector, F is the gridless EFGM overall force load column vector, F α is the column vector penalty term of the meshless EFGM global force load, and its submatrix expressions are as follows: In the formula, the strain matrix Elastic matrix of the jth anisotropic material Transformation matrix of the jth anisotropic material Shape function matrix m is the number of anisotropic materials, θ j is the orientation angle of the jth anisotropic material, α is the penalty factor of the penalty function method, is a given surface force, is the body force and displacement constraint judgment matrix on the design domain When the displacement constraint is applied in the x or y direction, the corresponding s1 and s2 are 1, otherwise they are 0; (11) The governing equations for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN are established as follows: Where w k is the neural network weight associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, b k is the neural network bias associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, n represents the number of hidden layers in the fully connected feedforward neural network FNN, The global stiffness moment of the meshless EFGM with the relative node density penalty is introduced. is the gridless displacement parameter column vector, F is the gridless EFGM overall force load column vector, K fα is the penalty term of the EFGM global stiffness matrix, F α is the penalty term of the meshless EFGM global force load column vector, φi is the shape function of the meshless EFGM, is the relative density of the j-th material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN, is the relative density of the jth material at the lth meshless EFGM calculation point, ng is the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on meshless EFGM and FNN, N is the number of nodes in the influence domain of the lth calculation point, and η j is the volume fraction of the jth anisotropic material, V0 is the total volume of the design domain, and s represents the number of meshless EFGM nodes in the design domain; (12) Solve the displacement value U of the multi-material structure and calculate the anisotropic multi-material structure flexibility C of the meshless EFGM and FNN Where, is the Young's modulus of the jth anisotropic material in the η direction in the material coordinate system, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, u l is the displacement matrix at the lth meshless EFGM calculation point, k l is the stiffness matrix at the lth meshless EFGM calculation point, is the flexibility of the jth anisotropic material at the lth meshless EFGM calculation point, ng represents the number of meshless EFGM calculation points in the design domain, and p is the penalty factor of the SIMP method for topology optimization; (13) Calculate the structure volume of the current training step and define the loss function Where, represents the current topology structural flexibility in the anisotropic multi-material topology optimization based on meshless EFGM and FNN, C0 represents the structural flexibility obtained by the first execution of training in the anisotropic multi-material topology optimization based on meshless EFGM and FNN, and V j represents the volume of the jth anisotropic material, V0 represents the total volume of the design domain, and v j represents the volume fraction of the jth anisotropic material, λ is the relaxation factor of the volume constraint in the topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN, and m represents the number of anisotropic material types; (14) Perform network training and calculate the derivative of the loss function with respect to the neural network parameters Update the network parameters using backpropagation, Where, Furthermore, the derivative of the structural flexibility with respect to the relative density of the meshless EFGM calculation points is: Therefore, equation (20) can be expressed as (15) Input the iteration termination condition. The iteration termination condition is that the infinite norm of the structural flexibility and its average value in ten consecutive iteration steps is less than 10 -3 If the termination condition is met, the iteration is terminated and the optimal topology structure of anisotropic multi-material structure topology optimization based on meshless EFGM and FNN is output. If not, the subsequent steps are continued and steps (9)-(15) are looped until the termination condition is met.

[0018] The following is a numerical example of the method of the present invention: See also Figure 3 This embodiment is a cantilever beam structure. Considering the process requirements of installation and use, the size is defined as 10m×5m, and the number of meshless EFGM nodes in the discretized design domain is 160×80. The left boundary of the structure is fixed, and the center position of the right boundary is applied with a downward size of F=1×10 4 N load. Define the Young's modulus of the five materials including the empty phase material Poisson's ratio of five materials When the five materials are isotropic, the Poisson's ratio factors Bt1 = Bt2 = Bt3 = Bt4 = Bt5 = 1, and the material orientation angles θ1 = θ2 = θ3 = θ4 = θ5 = 0°. When material four is anisotropic, its Poisson's ratio factor Bt4 = 3, and the material orientation angles θ4 = [0°, 30°, 60°, 90°, 120°, 150°]. The volume fractions of each solid material are defined to be equal, and the sum of the volume fractions is 0.4. The depth h of the neural network framework is 4, 6, 8, 10, 12, and 14, respectively. The number of neurons in each layer is 80, and the learning rate lr of the neural network framework is 0.001, 0.003, 0.005, 0.007, 0.009, and 0.011, respectively.

[0019] The specific implementation steps of the present invention for this example are as follows: (a) Input the design domain size and volume fraction η j , enter the Young's modulus of the anisotropic material Poisson's ratio Poisson's ratio factor Bt j , material orientation angle θ j , given the design domain boundary conditions and load size, the calculation point information in the design domain is obtained according to the meshless node information of the design domain, the background grid and the given boundary conditions; (b) According to the theory of element-free Galerkin method, the moving least squares method is used to construct the shape function to approximate the unknown field function. According to formula (5), the MLS approximate expression u(x) at x is h (x); (c) Calculate the support region radius r and the spline weight function ω(r) according to formula (7), and calculate the minimization functional J according to formula (6); (d) Calculate the unknown coefficient vector λ(x) according to formula (8); (e) Calculate the matrix A(x), B(x), the vector u and the vector N(x) composed of the MLS shape function without mesh nodes, and calculate the function u(x) according to formula (9); (f) Find the nodes within the influence domain of the calculation point according to the defined mesh-free support domain radius; (g) Calculation of the nodal stiffness matrix K of the meshless EFGM e Given the design domain boundary conditions and load size, the penalty function method is used to deal with various force and displacement boundary conditions. According to formulas (10) and (11), the stiffness matrix penalty term K is obtained. fα and force load penalty term F α ; (h) Determine the number of neural network layers, the number of neurons in each layer, the type of activation function, and the network learning rate, and build a fully connected feedforward neural network; (i) The meshless EFGM node coordinates are used as neural network input to predict the node relative density field of the multi-material structure; (j) The relative density of the meshless EFGM calculation points is obtained by interpolation based on the relative density of the nodes of the multi-material structure. The meshless EFGM global stiffness matrix of the design domain is assembled according to the relative density of the meshless EFGM calculation points, and the meshless EFGM discrete control equations of the structural statics problem are established according to formula (12); (k) Solve the displacement value U of the multi-material structure and calculate the system flexibility C according to formula (18); (1) Calculate the current topological structure volume V and define the loss function according to formula (19); (m) Perform network training and update network parameters using backpropagation; (n) Input the iteration termination condition. The iteration termination condition is that the infinite norm of the structural flexibility and its average value in ten consecutive iteration steps is less than 10 -3 ,If the termination condition is met, the iteration is terminated and the optimal topology structure of anisotropic multi-material structure topology optimization based on meshless EFGM and FNN is output. If not, the subsequent steps are continued and steps (i)-(n) are looped until the termination condition is met.

[0020] Figure 4-Figure 22 It is the optimal topology obtained by topology optimization of anisotropic multi-material structure based on meshless EFGM and FNN in this embodiment. Figure 4 The optimal topological structure obtained by using two materials (including a void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt5=1, and the material orientation angle θ1=θ5=0° in the embodiment of the present invention; Figure 5 The optimal topological structure obtained by using three materials (including a void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 6 The optimal topological structure obtained by using four materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt2=Bt3=Bt5=1, and the material orientation angles θ1=θ2=θ3=θ5=0° in the embodiment of the present invention; Figure 7 The optimal topological structure obtained by using four materials (including the void phase material) when the number of neural network layers h=4, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt2=Bt3=Bt5=1, and the material orientation angles θ1=θ2=θ3=θ5=0° in the embodiment of the present invention; Figure 8 The optimal topological structure obtained by using four materials (including the void phase material) when the number of neural network layers h=6, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt2=Bt3=Bt5=1, and the material orientation angles θ1=θ2=θ3=θ5=0° in the embodiment of the present invention; Figure 9 The optimal topological structure obtained by using four materials (including the void phase material) when the number of neural network layers h=10, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt2=Bt3=Bt5=1, and the material orientation angles θ1=θ2=θ3=θ5=0° in the embodiment of the present invention; Figure 10 The optimal topological structure obtained by using four materials (including the void phase material) when the number of neural network layers h=12, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt2=Bt3=Bt5=1, and the material orientation angles θ1=θ2=θ3=θ5=0° in the embodiment of the present invention; Figure 11The optimal topological structure obtained by using four materials (including the void phase material) when the number of neural network layers h=14, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt2=Bt3=Bt5=1, and the material orientation angles θ1=θ2=θ3=θ5=0° in the embodiment of the present invention; Figure 12 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.003, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 13 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.005, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 14 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.007, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 15 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.009, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 16 The optimal topological structure obtained by using three materials (including a void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.011, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 17 The optimal topological structure obtained by using three materials (including a void phase material) is as follows: the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, and the material orientation angles θ1=θ4=θ5=0° in the embodiment of the present invention; Figure 18 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=30° in the embodiment of the present invention; Figure 19The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=60° in the embodiment of the present invention; Figure 20 The optimal topological structure obtained by using three materials (including a void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, and the material orientation angles θ1=θ5=0° and θ4=90° is shown in the embodiment of the present invention; Figure 21 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=120° in the embodiment of the present invention; Figure 22 The optimal topological structure obtained by using three materials (including the void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factors Bt1=Bt5=1, Bt4=3, the material orientation angles θ1=θ5=0°, θ4=150° in the embodiment of the present invention; Figure 23 The optimal topological structure 3D printing model obtained by using two materials (including a void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt5=1, and the material orientation angle θ1=θ5=0° in the embodiment of the present invention; Figure 24 The optimal topological structure 3D printing model obtained by using three materials (including a void phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt5=1, and the material orientation angle θ1=θ2=θ5=0° in the embodiment of the present invention; Figure 25 This is the optimal topological structure 3D printing model obtained by using four materials (including the empty phase material) when the number of neural network layers h=8, the number of neurons in each layer n=80, the learning rate lr=0.001, the Poisson's ratio factor Bt1=Bt2=Bt3=Bt5=1, and the material direction angle θ1=θ2=θ3=θ5=0° in the embodiment of the present invention. Figure 4-Figure 25It can be seen that the optimal topological structure obtained by the anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN has clear and smooth boundaries, without numerical instability phenomena such as intermediate density and checkerboard, which is convenient for subsequent performance analysis and processing and manufacturing; there are certain differences in the optimal topological structures under different neural network depths, indicating that there are differences in their comprehensive performance. The fitting ability of the network framework can be controlled by adjusting the neural network depth to obtain a better topological structure; the learning rate lr mainly affects the convergence speed of the algorithm. Appropriate adjustment of the learning rate can accelerate convergence. The orientation angle θ of the anisotropic material j It will affect the topological flexibility value and the branching direction of the anisotropic material in the topological structure by adjusting the anisotropic material direction angle θ j The topological structure with specific mechanical properties is obtained. For less than and equal to five materials, the number of nodes of the meshless EFGM is a cantilever beam structure of 160×80, the network depth is 8 to 10, the learning rate lr ranges from 0.001 to 0.01, and the anisotropic material orientation angle θ j The values ​​range from 0° to 30° and 120° to 150°. For the same structural optimization problem, using multiple materials under the same network parameters can yield a richer range of topological designs. The number of material types can be selected based on actual engineering needs. This demonstrates that by adjusting these parameters, the optimal topology and performance of anisotropic multi-material structural topology optimization based on meshless EFGM and FNN can be controlled, demonstrating promising theoretical research and engineering application value.

[0021] Although the present invention has been described in detail with reference to this embodiment, the above description does not limit the scope of protection of the present invention. Any modifications and improvements based on the concept of the present invention are deemed to be within the scope of protection of the present invention.

Claims

1. Anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN, characterized by The following steps are involved: (1) According to the design requirements of the actual engineering structure, determine the initial design domain of the structure and the volume fraction η of each anisotropic material in the multi-material structure. j , enter the Young's modulus of each anisotropic material Poisson's ratio Shear modulus Poisson's ratio factor Bt j , anisotropic material direction angle θ j , given the design domain boundary conditions and load size, the calculation point information in the design domain is obtained according to the meshless node information of the design domain, the background grid and the given boundary conditions; (2) Calculate the structural stiffness matrix without introducing node relative density penalty based on meshless EFGM and apply boundary conditions: (a) Solve the elastic matrices of various materials according to the mechanical properties of anisotropic materials Where T j is the transformation matrix of the anisotropic multi-material between the meshless EFGM and FNN, where θ j is the orientation angle of the jth anisotropic material based on the meshless EFGM and FNN, as well as They are the Young's modulus and Poisson's ratio of the jth anisotropic material in the ξ and η directions in the material coordinate system, and satisfy the relationship Bt j is the Poisson's ratio factor of the jth anisotropic material, is the shear elastic modulus of the jth anisotropic material; (b) The shape function φ of the meshless EFGM is constructed using the moving least squares approximation i , and integrate at the calculation point to calculate the meshless EFGM anisotropic multi-material structure stiffness matrix without introducing node relative density penalty Where the strain matrix Elastic matrix of the jth anisotropic material Where, (c) Determine the overall force load vector F of the meshless EFGM in the design domain according to the magnitude of the force load applied in the design domain. Where the shape function matrix N(x) is the vector of MLS shape functions corresponding to the grid-free nodes in the neighborhood of the calculation point x. is the force matrix on the given boundary Γ, is the force on the design domain Ω; (d) According to the given force and displacement boundary conditions, the penalty function method is used to process various force and displacement boundary conditions to obtain the penalty term K of the meshless EFGM overall force stiffness matrix fα and the EFGM overall force load vector penalty term F α ,in, Where α is the penalty factor of the penalty function method, and the displacement constraint judgment matrix When the displacement constraint is applied in the x or y direction, the corresponding s1 and s2 are 1, otherwise they are 0. is the displacement constraint matrix; (3) Predict the relative density field of the gridless EFGM nodes of the topological structure and calculate the displacement of the topological structure: (a) Determine the number of neural network layers, the number of neurons in each layer, the activation function type, the network learning rate, build a fully connected feedforward neural network, initialize the weights and biases of the neural network, and the neural network parameters It is composed of the weight matrix w and bias matrix b of all layers Where w=[w1,w2,...,w n ],b=[b1,b2,...,b n ],w n represents the weight associated with the nth layer of neurons, b n represents the bias associated with the nth layer of neurons; (b) The meshless EFGM node coordinates are used as the input of the fully connected feedforward neural network to predict the meshless EFGM node relative density field of the anisotropic multi-material structure based on the meshless EFGM and FNN. In the formula is the matrix composed of the density of each node s represents the number of meshless EFGM nodes in the design domain, m represents the number of anisotropic material types in the design domain, x represents the meshless EFGM node coordinate matrix x, BN represents batch normalization of the input information, σ represents the ReLU activation operation performed on the input data, and Softmax represents the Softmax activation operation performed on the input data; (c) The relative density of the meshless EFGM of the multi-material structure at the calculation point is obtained by interpolation based on the predicted relative density of the meshless EFGM nodes. Where, represents the relative density of the jth material at the lth meshless EFGM calculation point, N represents the number of meshless EFGM nodes in the influence domain of the lth calculation point, represents the relative density of the jth material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN; (d) the meshless EFGM overall stiffness matrix of the design domain assembled according to the relative density of the meshless EFGM calculation points ng represents the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on the meshless EFGM and FNN, and p is the penalty factor of the topology optimization SIMP method; (e) The static discrete governing equations of the meshless EFGM are established to solve the displacement parameter values ​​of the meshless EFGM nodes in the design domain. The displacement value U of each meshless EFGM node is solved according to the displacement parameter value of the meshless EFGM node in the design domain; (4) Calculate the flexibility of anisotropic multi-material structures based on meshless EFGM and FNN and update the network parameters of FNN by back propagation: (a) Calculate the flexibility of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, u l is the displacement matrix at the lth meshless EFGM calculation point, k l is the stiffness matrix at the lth meshless EFGM calculation point, is the flexibility of the jth anisotropic material at the lth meshless EFGM calculation point, and ng represents the number of meshless EFGM calculation points in the design domain; (b) Calculation of the volume of anisotropic multi-material topology based on meshless EFGM and FNN Defining the loss function of a fully connected feedforward neural network (FNN) for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the topological flexibility of the current training step in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, C0 represents the topological flexibility obtained by the first execution of training in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, V0 represents the total volume of the design domain, and v j represents the volume fraction of the jth anisotropic material, and λ is the relaxation factor of the volume constraint in the topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN; (c) FNN network training is performed, and the derivatives of the loss function with respect to the neural network parameters are calculated using back propagation. And update the network parameters, (5) The governing equations for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN are established as follows: Where w k is the neural network weight associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, b k is the neural network bias associated with the k-th layer of neurons in the fully connected feedforward neural network FNN, n represents the number of hidden layers in the fully connected feedforward neural network FNN, The global stiffness moment of the meshless EFGM with the relative node density penalty is introduced. is the gridless displacement parameter column vector, F is the gridless EFGM overall force load column vector, K fα is the penalty term of the EFGM global stiffness matrix, F α is the penalty term of the meshless EFGM global force load column vector, φ i is the shape function of the meshless EFGM, is the relative density of the j-th material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN, is the relative density of the jth material at the lth meshless EFGM calculation point, ng is the number of meshless EFGM calculation points in the design domain, m is the number of material types of the anisotropic multi-material structure based on meshless EFGM and FNN, N is the number of nodes in the influence domain of the lth calculation point, and η j is the volume fraction of the jth anisotropic material, V0 is the total volume of the design domain, and s represents the number of meshless EFGM nodes in the design domain; (6) Input the iteration termination condition. If the termination condition is met, the iteration is terminated, and the optimal topology structure of the anisotropic multi-material structure topology optimization based on the meshless EFGM and FNN is output according to the relative density value of each meshless node. If it is not met, continue to execute the subsequent steps and loop steps (3)-(4) until the termination condition is met.

2. The anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN according to claim 1 is characterized in that Determine the number of neural network layers, the number of neurons in each layer, the activation function type, the network learning rate, build a fully connected feedforward neural network, initialize the weights and biases of the neural network, and the neural network parameters It is composed of the weight matrix w and bias matrix b of all layers Where w=[w1,w2,...,w n ],b=[b1,b2,...,b n ],w n represents the weight associated with the nth layer of neurons, b n Represents the bias associated with the nth layer of neurons; the meshless EFGM node coordinates are used as the input of a fully connected feedforward neural network to predict the node relative density field of anisotropic multi-material structures based on the meshless EFGM and FNN. In the formula is the matrix composed of the density of each node s represents the number of meshless EFGM nodes in the design domain, m represents the number of anisotropic material types in the design domain, x represents the meshless EFGM node coordinate matrix x, BN represents batch normalization of the input information, σ represents the ReLU activation operation on the input data, and Softmax represents the Softmax activation operation on the input data. The relative density of the meshless EFGM of the multi-material structure at the calculation point is obtained by interpolation based on the predicted relative density of the meshless EFGM nodes. Where, represents the relative density of the jth material at the lth meshless EFGM calculation point, N represents the number of meshless EFGM nodes in the influence domain of the lth calculation point, It represents the relative density of the j-th material at the i-th meshless EFGM node predicted by the fully connected feedforward neural network FNN.

3. The anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN according to claim 1 is characterized in that The structural flexibility is calculated based on the displacement value U of the meshless EFGM node calculated in step (3) of claim 1, and the flexibility of the anisotropic multi-material structure based on the meshless EFGM and FNN is calculated. Where, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, u l is the displacement matrix at the lth meshless EFGM calculation point, k l is the stiffness matrix at the lth meshless EFGM calculation point, is the flexibility of the jth anisotropic material at the lth meshless EFGM calculation point, ng represents the number of meshless EFGM calculation points in the design domain; calculate the volume of the anisotropic multi-material topology based on the meshless EFGM and FNN Defining the loss function of a fully connected feedforward neural network (FNN) for topology optimization of anisotropic multi-material structures based on meshless EFGM and FNN Where, represents the topological flexibility of the current training step in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, C0 represents the topological flexibility obtained by the first execution of training in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN, represents the relative density of the jth anisotropic material at the lth meshless EFGM at the calculation point, V0 represents the total volume of the design domain, and v j represents the volume fraction of the jth anisotropic material, λ is the relaxation factor of the volume constraint in the anisotropic multi-material structure topology optimization based on meshless EFGM and FNN; performs FNN network training and uses backpropagation to calculate the derivative of the loss function with respect to the neural network parameters And update the network parameters, 4. The anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN according to claim 1 is characterized in that The network depth of the neural network model and the number of neurons in each layer of the network need to be determined according to the number of nodes of the meshless EFGM. For a cantilever beam structure with less than or equal to five materials and the number of nodes of the meshless EFGM is 160×80, the network depth ranges from 8 to 10, the number of neurons in each layer of the network is 80, and the learning rate lr ranges from 0.001 to 0.

01.

5. The anisotropic multi-material structure topology optimization method based on meshless EFGM and FNN according to claim 1 is characterized in that Orientation angle θ of anisotropic multi-materials based on meshless EFGM and FNN j It will affect the mechanical properties of the topological structure. For the multi-material cantilever beam structure, the orientation angle θ of the anisotropic multi-material based on the meshless EFGM and FNN is j The value range is 0°~30° and 120°~150°.

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