Multi-material structure topological optimization method based on meshless EFGM and style migration

Abstract

The invention discloses a meshless EFGM and style migration-based multi-material structure topological optimization method, which comprises the following steps of: (1) inputting material attributes such as Young modulus, shear modulus and Poisson's ratio of a structure, and discretizing a design domain by using meshless nodes; (2) transforming the designed meshless density field to obtain a physical density field, and performing length scale filtering; (3) an alternating active phase algorithm is executed, and the displacement field, the system flexibility and the related sensitivity of the structure are solved based on the meshless EFGM; (4) executing a neural style migration program to calculate the difference between the current topological structure and the reference style image, and calculating the derivative of the physical density field of the topological structure to the network parameter; and (5) updating a design variable by using a moving asymptote algorithm, and judging whether an optimization loop is converged or not. According to the method, based on meshless EFGM and style migration multi-material structure topological optimization, topological structures with different artistic styles can be designed, and the method has engineering application value.

Description

Technical Field

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

[0002] Topology optimization can effectively reduce production costs while ensuring the mechanical properties of the structure, and has a wide range of application scenarios in the engineering field. In the past, single materials were mostly used in topology optimization, and the designed structure was difficult to withstand complex external loads. Multi-material topology design can explore the optimization possibilities of the structure in a wider design space, and achieve a structural design with stronger comprehensive performance by flexibly selecting a variety of materials with different properties and accurately arranging them in the structure. This design method can remove inefficient materials and better utilize the characteristics of each material, thereby significantly improving the mechanical properties, stiffness and durability of the structure without increasing weight or cost. In addition, multi-material structural topology optimization also has higher design freedom and better flexibility. Through the combination and optimization of different materials, it can fully tap the potential of materials and achieve further breakthroughs in structural performance.

[0003] At present, mesh-based numerical calculation methods are usually used in structural topology optimization to achieve structural performance analysis, such as Finite Element Method (FEM), Finite Volume Method (FVM) and Finite Difference Method (FDM). Such mesh-based calculation methods are easily affected by deformed meshes, resulting in numerical instability, which is reflected in topology optimization as checkerboard phenomenon, intermediate density, and difficulty in objective function convergence. Meshless methods construct shape functions by influencing nodes in the domain, so that they can generate higher-order shape functions more flexibly when dealing with complex geometric shapes and boundary conditions. This higher-order shape function has higher approximation accuracy, which helps to more accurately capture subtle changes and complex behaviors of the structure without relying on fixed mesh division, and has better numerical stability in numerical calculations. Element-free Galerkin Method (EFGM) is a relatively mature meshless method that can handle solid mechanics problems well. The topology optimization method based on meshless EFGM relies on the original sensitivity of the objective function, effectively avoids the checkerboard phenomenon, can obtain clear topological boundaries, and further promote the engineering application of topology optimization.

[0004] Traditional topology optimization methods focus on the mechanical properties of the structure. For various designs that consider aesthetics, manual post-processing or simply changing the topology design parameters is usually used. This method that relies on manual post-processing is time-consuming and labor-intensive, may not be effective, and it is difficult to integrate existing design styles and elements into topology optimization. Style transfer technology (Neural Style Transfer, NST) is an image processing method based on deep learning. It applies the style features of one image to another image through a convolutional neural network. While maintaining the structure of the content image, it integrates the texture and color features of the style image into the content image to generate an image with a specified style. Topology optimization aims to improve the performance of the structure by optimizing the material distribution, and style transfer technology can transform a specific design style into an optimization target. This combination allows designers to introduce visual aesthetic guidance in the process of topology optimization, so that the final structure not only achieves the best performance, but also has a unique appearance and style. The introduction of style transfer technology provides a new perspective and method for topology optimization, promoting the dual progress of engineering design in aesthetics and function.

[0005] In traditional topology optimization, since the optimization algorithm pursues the ultimate material distribution, the generated structure may contain too small features or unrealistic geometric shapes. These structures are often difficult to achieve in actual manufacturing, and too small structural features may cause local stress concentration or insufficient material, thus affecting the overall performance and safety of the structure. Length scale control technology is a strategy used to optimize topological structure design. Its core purpose is to control the minimum size of structural features during the optimization process. This technology introduces length scale parameters to ensure that the structure generated during the optimization process has appropriate geometric features and avoids too small structural details. It can effectively solve the common problem of structural over-refinement in topology optimization, thereby improving the practicality and manufacturability of structural design. Summary of the invention

[0006] With the continuous development of science and technology, people pay attention to the structural performance of products while also increasing their demand for aesthetic features, from traditional mechanical mechanism design (aerospace, transportation, electronic devices, etc.) to architectural design and other fields. Traditional topology optimization methods can ensure good mechanical properties of the structure, but the aesthetic features of the structure need to rely on manual post-processing, which is time-consuming and labor-intensive and may not be effective. In order to solve the numerical instability phenomena such as sawtooth, checkerboard and intermediate density that are prone to occur in the topological structure obtained based on the common topology optimization method, the topological structure has a good artistic atmosphere at the same time. The present invention proposes a multi-material structure topology optimization method based on meshless EFGM and NST, using meshless EFGM node discretization design domain, meshless EFGM node relative density as design variable, volume fraction as constraint condition, and the topological result obtained by meshless EFGM is converted into an image as the input of the VGG-19 network model, and the difference between it and the reference style image is calculated. Then the topological structure flexibility and neural network loss function obtained by the meshless EFGM method are used as optimization targets, and a topological optimization mathematical model of multi-material structure based on meshless EFGM and NST is established. A computer program is written to write an algorithm to perform topology optimization design under different content and style weight coefficients and length scale control parameters, and output its optimal topology structure.

[0007] The technical solution adopted by the present invention to solve its technical problems is: a multi-material structure topology optimization method based on meshless EFGM and NST, using the static control equation of the meshless EFGM discretized structure. In combination with the style transfer technology, the topological structure obtained by the meshless EFGM is converted into an image as the network input of the style transfer, and the difference between it and the reference style image is calculated as the network loss function, and the flexibility of the topological structure and the loss function are minimized to realize the topological mechanism design with artistic style. The artistic style of the topological structure is controlled by adjusting the weight coefficient of the content and style in the style transfer, and the minimum scale in the topological structure is controlled by adjusting the scale control parameter.

[0008] 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, material volume constraint η, material type q, meshless EFGM initial design density field ρ, topological structure content and style weight coefficient w c and w s , set the elastic modulus E1 and Poisson's ratio v of the material 12 , shear modulus G 12 , Poisson's ratio factor Bt, material orientation angle θ, given the design domain boundary conditions and load size, according to the meshless EFGM node information of the design domain, the background grid and the given boundary conditions, the Gaussian point information of the design domain is obtained; (2) Calculate the system stiffness matrix and load vector based on the meshless EFGM: (a) Determine the meshless EFGM overall force load vector of the design domain according to the magnitude of the force load applied in the design domain; (b) According to the given force and displacement boundary conditions, process various force and displacement boundary conditions, and use the penalty function method to process the displacement boundary to obtain the meshless EFGM overall force stiffness matrix penalty term and the meshless EFGM overall force load vector penalty term; (c) Calculate the material's elastic matrix based on the material's mechanical properties in, as well as are the elastic modulus and Poisson's ratio of the i-th material in the x and y directions in the material coordinate system, respectively, and satisfy the relationship The meshless EFGM global stiffness matrix of the design domain is assembled according to the meshless EFGM node relative density distribution function, and the meshless EFGM discrete control equations of the structural statics problem are established; (3) Length scale control of the design density field: (a) Filter the gridless EFGM design density field ρ to obtain the gridless EFGM filtered density field In the formula v l represents the volume of the lth meshless EFGM node, and the weight coefficient ||x k -x l || represents the distance between the kth gridless EFGM node and the lth gridless EFGM node, d fil is the filter diameter; (b) the physical density field is then calculated based on the filter density field β represents the projection steepness, μ represents the threshold; (c) Calculate the average density of the physical density field in the scale control domain in w (k,l) =H(d min -2||x k -x l ||), d min represents the length scale controlling the diameter, H(·) represents the Heaviside function; (d) the sum of the weighted density is calculated based on the average density and the approximate maximum mean density Where p1 is the exponential factor; (e) Then calculate the scale control aggregation function in p1 is the exponential factor, N is the number of meshless EFGM nodes in the design domain, and ξ is the relaxation factor for scale control; (4) Calculate the difference between the topological structure obtained by the current gridless EFGM and the reference image based on the VGG-19 network framework: (a) Convert the physical density field based on the gridless EFGM The constructed topological structure is converted into an RGB image image1; (b) a reference image imgRef is defined for extracting the image style; (c) the VGG-19 network is used to extract the content and style features of the reference image imgRef; (d) the VGG-19 network is used to extract the pixel features of the image image1; (e) the difference between the pixel features of the image image1 and the extracted content and style pixel features of the reference image imgRef is calculated, which are the content loss L c and style loss L s ; (f) Multiply the content loss and style loss by the corresponding weight coefficient w c and w s Get the total loss of the network model Loss = w c L c +w s L s ; (g) Calculate the physical density field Gradients of the neural network parameters w and b And output parameters; (5) Calculate the density distribution of multi-material structures based on the alternating active-phase algorithm (AAPA): (a) Define the fixed phase a and the activated phase b of the material, where the value of the a-phase material is traversed in the range of [1, q], and the b-phase material is traversed in the sub-cycle, with a range of [a+1, q]; (b) According to the meshless EFGM global force load vector determined in step 2(a) and the meshless EFGM global force stiffness matrix penalty term and meshless EFGM global force load vector penalty term obtained by the penalty function method in step 2(b), solve the displacement parameter value of the meshless EFGM node in the design domain; solve the displacement value of each meshless node and the system flexibility C based on the displacement parameter value of the meshless EFGM node in the design domain; (c) Calculate the sensitivity C of the flexibility corresponding to the two materials a and b to the design variables a and C b , calculate the flexibility difference C1 = C a -C b And its filtering density field for a-phase material sensitivity dc, and volume sensitivity dv a (d) Calculation Filtering density field for a-phase material The derivative of (e) Calculate the difference V between the current volume of the a-phase material and the target volume diff_a , V diff_a As a volume constraint; the VGG-19 network output parameters Multiply by the weight factor w L Get the style constraint S diff_a(f) Update the physical density field of phase a material using the moving asymptotes algorithm (MMA) The system flexibility C is taken as the optimization target, and the volume constraint V diff_a and style constraint S diff_a As a constraint function; (g) Calculate the relative density ρ of the physical field of phase b material b ; (6) Based on the meshless EFGM and style transfer related theories, the mathematical model of topology optimization of multi-material structures is established as follows: In the formula, represents the density of the jth material at the i-th meshless EFGM node, F is the meshless EFGM global force load column vector, and F α is the meshless EFGM total force load column vector penalty term, U is the meshless EFGM displacement column vector, To introduce the relative density value into the meshless EFGM global stiffness matrix, E0 is the elastic modulus of the solid material, E min In order to prevent the stiffness matrix from being singular, a smaller value of Young's modulus is given, is the global stiffness matrix penalty term of the meshless EFGM, H(·) is the Heaviside function, φ I is the meshless EFGM interpolation shape function, is the density of the jth material at the ith Gaussian point in the meshless EFGM, V represents the material volume, η (j) represents the volume fraction of the jth material, L diff,h is the loss function of the VGG-19 network model, δ is the relaxation factor used to judge the convergence of the loss function, and g m is the scale control constraint aggregation term, ξ is the relaxation factor for the scale control constraint, They represent the design density, filter density, and physical density of the j-th material at the i-th meshless EFGM node, respectively; (7) Define the iteration termination condition. If the termination condition is met, the iteration is terminated, and the optimal topological structure of multi-materials with scale control based on meshless EFGM and style transfer is output according to the physical field density value of each meshless EFGM node. If it is not met, continue to execute the subsequent steps and loop step (5) until the termination condition is met.

[0009] The beneficial effects of the present invention are as follows: based on the meshless Galerkin method and style transfer technology, the present invention effectively eliminates the numerical instability problems such as jaggedness, checkerboard and intermediate density that are prone to occur in the topological structure obtained based on the common topological optimization method; at the same time, a topological structure with a specific style can be obtained; the present invention adopts a length scale control method, and obtains a topological structure with a specified size constraint by filtering the node density of the meshless EFGM, thereby solving the common problem of excessive structural refinement in topological optimization, thereby improving the practicality and manufacturability of structural design; the topological structure obtained by the present invention has a clear outline and is easy to process and manufacture; the present invention can achieve the optimal distribution of multiple materials in the topological structure according to different design requirements, provide an effective solution for the topological structure design under complex working conditions, and has important engineering application value; the present invention controls the artistic style of the topological result by adjusting the style weight coefficient and the content weight coefficient in the style transfer, and can realize different types of topological structure designs according to engineering requirements, and the operation is simple. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] The present invention is further described in detail below with reference to the accompanying drawings and embodiments.

[0011] Figure 1 This is the topology optimization design flow chart of the multi-material structure based on meshless EFGM and style transfer NST of the present invention. Figure 2 Schematic diagram of the structural design area and non-design area in an embodiment of the present invention Figure 3 Schematic diagram of boundary conditions and force loads in an embodiment of the present invention Figure 4 It is the VGG-19 network model framework used in the embodiment of the present invention Figure 5 is the reference style image selected in the embodiment of the present invention Figure 6 This is the optimal topology obtained when NST is not used in the embodiment of the present invention. Figure 7 This is the optimal topological structure obtained when the style loss selects the convolution layer Cov5_1 in the embodiment of the present invention. Figure 8 This is the optimal topological structure obtained when the style loss selects convolutional layers Cov4_1 and Cov5_1 in the embodiment of the present invention. Fig. 9 This is the optimal topological structure obtained when the style loss selects convolutional layers Cov3_1, Cov4_1, and Cov5_1 in the embodiment of the present invention. Fig.10 It is the optimal topological structure obtained when the style loss selects the convolution layers Cov2_1, Cov3_1, Cov4_1, and Cov5_1 in the embodiment of the present invention. Fig.11 It is the optimal topological structure obtained when the style loss selects the convolution layers Cov1_1, Cov2_1, Cov3_1, Cov4_1, and Cov5_1 in the embodiment of the present invention. Fig.12 It is the optimal topological structure obtained when the content loss selects the convolution layer Cov2_1 in the embodiment of the present invention. Fig.13 This is the optimal topological structure obtained when the content loss selects the convolution layer Cov3_1 in the embodiment of the present invention. Fig.14 This is the optimal topological structure obtained when the content loss selects the convolution layer Cov4_1 in the embodiment of the present invention. Fig.15 This is the optimal topological structure obtained when the content loss selects the convolution layer Cov5_1 in the embodiment of the present invention. Fig.16 This is the optimal topological structure obtained when the content loss selects the convolution layer Cov5_2 in the embodiment of the present invention. Fig.17 is the style loss weight w in the embodiment of the present invention s =1.0, content loss weight w c = 0 when the optimal topology is obtained Fig.18 is the style loss weight w in the embodiment of the present invention s =0.75, content loss weight w c = 0.25 when the optimal topology is obtained Fig.19 is the style loss weight w in the embodiment of the present invention s =0.5, content loss weight w c = 0.5 when the optimal topology is obtained Fig. 20 is the style loss weight w in the embodiment of the present invention s =0.25, content loss weight w c = 0.75 when the optimal topology is obtained Fig.21 is the style loss weight w in the embodiment of the present invention s =0, content loss weight w c =1.0 to obtain the optimal topology Fig. 22 This is the optimal topological structure obtained when the minimum scale control is used instead of NST in the embodiment of the present invention. Fig.23 is the style loss weight w in the embodiment of the present invention s =0.5, content loss weight w c = 0.5 when the optimal topology is obtained using the minimum scale control DETAILED DESCRIPTION

[0012] See also Figure 1 The length-scale controlled topology optimization method of multi-material structures based on meshless EFGM and NST mainly includes the following steps: First, determine the elastic modulus E1 and Poisson's ratio v of the material 12 , shear modulus G 12 and material volume fraction η. According to Hooke's law, the relationship between material stress and strain is: In the formula, Q (i) is the elastic matrix of the ith material, Q 33 =G 12 , Among them, E1, E2 and v 12 、v 21 They are the elastic modulus and Poisson's ratio in the x and y directions in the material coordinate system, respectively, and satisfy the relationship G 12 is the shear elastic modulus.

[0013] Discretize the design domain into a series of nodes and calculate the Gaussian point information in the design domain. Find the meshless EFGM nodes in the Gaussian point influence domain and assemble the stiffness matrix. Establish the meshless EFGM discrete control equations for the linear elastic problem of multi-material structures, and use the penalty function method to apply force and displacement boundary conditions:

[0014] Execute the scale control procedure: (a) Filter the meshless EFGM design density field ρ to obtain the filtered density field In the formula v l represents the volume of the lth meshless EFGM node, and the weight coefficient ||x k -x l || represents the distance between the kth gridless EFGM node and the lth gridless EFGM node, d fil is the filter diameter; (b) the physical density field is then calculated based on the filter density field β represents the projection steepness, μ represents the threshold; (c) Calculate the average density of the physical density field in the scale control domain in w (k,l) =H(d min -2||x k -x l ||), d minrepresents the length scale controlling the diameter, H(·) represents the Heaviside function; (d) the sum of the weighted density is calculated based on the average density and the approximate maximum mean density p1 is the exponential factor (e) and then the scale control aggregation function is calculated in p1 is the exponential factor, N is the number of meshfree EFGM nodes in the design domain, and ξ is the relaxation factor for scale control.

[0015] Execute the alternating active phase algorithm to solve the structural displacement, calculate the system flexibility, the sensitivity of the a-phase material, and the volume sensitivity of the a-phase material. Convert the current topological physical density field into an RGB image as the input of the VGG-19 network model to extract image features; use the VGG-19 network model to extract the content and style features of the reference image; calculate the difference between the image features of the topological structure and the content and style features of the reference image, respectively, as the content loss term and style loss term of the loss function; update the network parameters through back propagation, and calculate the physical density field The derivatives of the network parameters And output.

[0016] Calculating style transfer network parameters Sensitivity to a-phase materials The MMA algorithm is used to update the density of the a-phase material, with the system flexibility C as the optimization target, the volume of the a-phase material and the weighted style transfer loss function as optimization constraints. The density of the b-phase material is calculated based on the total volume of the a- and b-phase materials.

[0017] Finally, a topology optimization mathematical model of multi-material structure based on meshless EFGM and NST is established: In the formula, represents the density of the jth material at the i-th meshless EFGM node, F is the meshless EFGM global force load column vector, and F α is the meshless EFGM total force load column vector penalty term, U is the meshless EFGM displacement column vector, To introduce the relative density value into the meshless EFGM global stiffness matrix, E0 is the elastic modulus of the solid material, E min In order to prevent the stiffness matrix from being singular, a smaller value of Young's modulus is given, is the global stiffness matrix penalty term of the meshless EFGM, H(·) is the Heaviside function, φ I is the meshless EFGM interpolation shape function, is the density of the jth material at the ith Gaussian point in the meshless EFGM, V represents the material volume, η(j) represents the volume fraction of the jth material, L diff,h is the loss function of the VGG-19 network model, δ is the relaxation factor used to judge the convergence of the loss function, and g m is the scale control constraint aggregation term, ξ is the relaxation factor for the scale control constraint, They represent the design density, filter density and physical density of the j-th material at the i-th meshless EFGM node, respectively.

[0018] See also Figure 1 , the specific steps of the length-scale controlled topology optimization method for multi-material structures based on meshless EFGM and NST are as follows: (1) According to the design requirements of the actual engineering structure, determine the initial design domain of the structure, the elastic modulus E1 of the material, and the Poisson's ratio v 12 , shear modulus G 12 and material volume fraction η and other properties. Given the boundary conditions and load size of the design domain, the Gaussian point information in the design domain is obtained according to the meshless EFGM node information, background mesh and given boundary conditions of the design domain; (2) Find the distance d of the ninth meshless EFGM node closest to the Gaussian point max , d max Multiply by the scaling factor as the radius of the Gaussian point influence domain d = d max ×s k ; (3) According to the theory of the element-free Galerkin method, the moving least squares (MLS) 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 In the formula, 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 set of unknown coefficient vectors consisting of functions of x.

[0019] The unknown coefficient vector λ(x) is obtained by minimizing the functional J, which is In the formula, u i is the function value of the desired function u(x) at the mesh-free node in the neighborhood of the calculation point x, NP is the number of mesh-free nodes in the neighborhood of the calculation point x, ω(x i ) is the weight function, and the cubic spline weight function is selected. Its specific expression is In the formula, r = || xx i || / d.

[0020] Therefore, λ(x) is λ(x)=A -1 (x)B(x)u (8) In the formula, B(x)=[ω1(x)a(x1)ω2(x)a(x2)···ω NP (x)a(x NP )],u=[u1 u2 u3…u NP ] T .

[0021] Substituting formula (8) into formula (5), we can obtain: 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).

[0022] Based on the meshless EFGM, the displacement field of anisotropic structures is solved. The meshless EFGM discretization governing equations of the orthotropic structural statics problem are summarized as follows: In the formula, is the meshless EFGM global force stiffness matrix, K fα is the penalty term of the meshless EFGM global force stiffness matrix, is the meshless EFGM displacement column vector, F is the meshless EFGM global force load column vector, F α is the penalty term of the gridless EFGM global force load column vector, and its submatrix expressions are as follows: In the formula, α is the penalty factor size, is a given surface force, is the body force on the design domain. When displacement constraints are applied in the x or y direction, the corresponding s1 and s2 are 1, otherwise they are 0.

[0023] The design density field is transformed to achieve topological structure length scale control. The main steps are as follows: (3.1) Transform the design density field ρ to obtain the filtered density field and physical density field In the formula, v l represents the volume of the lth meshless EFGM node, and the weight coefficient ||x k -x l || represents the distance between the kth gridless EFGM node and the lth gridless EFGM node, d fil is the filter diameter; (3.2) Then the physical density field is calculated based on the filtered density field: In the formula, β represents the projection steepness, μ represents the threshold; (3.3) Calculate the average density of the physical density field in the scale control domain In the formula, w (k,l) =H(d min -2||x k -x l ||), d min represents the length scale controlling diameter, H(·) represents the Heaviside function; (3.4) Calculate the weighted density and and the approximate maximum mean density (3.5) Then calculate the scale control aggregation function In the formula, p1 is the exponential factor, N is the number of meshfree EFGM nodes in the design domain, and ξ is the relaxation factor for scale control.

[0024] Execute the multi-material AAPA program and solve the multi-material structure density field based on meshless EFGM and NST. The detailed steps are as follows: (3.6) Determine the maximum number of iterations and convergence conditions, and execute the two-phase competition subroutine, with the material fixed phase and activation term being phases a and b respectively; (3.7) Determine the meshless EFGM global force load vector of the design domain according to the magnitude of the force load applied in the design domain; (3.8) According to the given force and displacement boundary conditions, various force and displacement boundary conditions are processed, wherein the penalty function method is used to process the displacement boundary, and the meshless EFGM global force stiffness matrix penalty term and the meshless EFGM global force load vector penalty term are obtained; (3.9) Solve the elastic matrix Q of the structure according to the mechanical properties of the material, assemble the meshless EFGM global stiffness matrix of the design domain, establish the meshless EFGM discrete control equation of the structural statics problem, combine the meshless EFGM global force load vector determined in step (3.7) and the meshless EFGM global force stiffness matrix penalty term and meshless EFGM global force load vector penalty term obtained by the penalty function method in step (3.8), and solve the displacement parameter value of the meshless EFGM node in the design domain; (3.10) According to the displacement parameter value of the meshless EFGM node in the design domain, the displacement value of each meshless EFGM node is solved to calculate the system flexibility C; (3.11) Calculate the flexibility difference C of the two phase materials a and b diff and the corresponding derivative difference dc diff ; Calculate dc diff Sensitivity dc to the initial design density field of the a-phase material, the difference between the volume of the a-phase material and the target volume V diff and volume sensitivity dv; (4) Execute the style transfer subroutine to calculate the difference between the current topology and the reference image. The main steps are as follows: (4.1) Convert the current topological physical density field into an RGB image as the input of the VGG-19 network model to extract image features; (4.2) Use the VGG-19 network model to extract the content and style features of the reference image; Calculate the difference between the image features of the topological structure and the content and style features of the reference image, which are used as the content loss term and style loss term of the loss function respectively; In the formula, and, (4.3) Update the network parameters through back propagation, (4.4) Calculation of physical density field The derivatives of the network parameters And calculate the style transfer network parameters Sensitivity to a-phase materials (5) Use the MMA algorithm to update the design variables. The main steps are as follows: (5.1) Update the density of phase a material, the system flexibility C is used as the optimization target, and the volume constraint V of phase a material is used as the optimization target. diff And the weighted style transfer loss function L diff =w l L diff,k (T k ,R) as optimization constraints; (5.2) Calculate the density of phase b material based on the total volume of phase a and phase b materials.

[0025] Determine whether to exit the calculation program based on the convergence conditions.

[0026] The following is an example of the application of the method of the present invention to engineering practice: See also Figure 3 , this embodiment is a support structure. To facilitate load application, it is divided into a design area and a non-design area. The non-design area is 0.05m high and 5m wide, and the non-design area is 4.95m high and 5m wide. The entire geometric structure is discretized using 22,500 meshless nodes. The middle position of the lower boundary of the structure is fixed at 0.5m, and the upper boundary is distributed with a uniform load q=1×10 4 N / m. The structure is composed of two materials, and the principal elastic modulus of the solid material is defined as E1 = [14, 10, 1 × 10 -9 ]Gpa, shear elastic modulus G 12 =E / 2(1+v 12 ), the volume fraction is η = [0.15, 0.15, 0.7], the main Poisson's ratio v 12 =[0.3,0.3,0.3], Poisson's ratio factor Bt = [1,1,1]. The style reference image is a tree-like structure, such as Figure 5 As shown. The style loss convolutional layers are Cov5_1, Cov45_1, Cov345_1, Cov2345_1, Cov12345_1; the content loss convolutional layers are Cov2_1, Cov3_1, Cov4_1, Cov5_1, Cov5_2; the style loss weight coefficient w s Take 1.0, 0.75, 0.5, 0.25, and 0 respectively.

[0027] The specific implementation steps of the present invention for this example are as follows: (a) Input the design domain size, volume constraint η, material elastic modulus E1, and Poisson’s ratio v 12 , shear modulus G 12 , Poisson's ratio factor Bt, given the boundary conditions and load size of the design domain, according to the meshless EFGM node information of the design domain, the background grid and the given boundary conditions, obtain the Gaussian point information of the design domain and the meshless EFGM nodes in the influence domain; (b) Step by step, search for the meshless EFGM nodes in the influence domain of each meshless Gaussian point in the design domain and calculate the MLS shape function of the corresponding meshless node; (c) Solve the elastic matrix Q of the anisotropic structure according to the mechanical properties of the material, and assemble the meshless EFGM global force stiffness matrix of the design domain according to the relative density distribution function; (d) Determine the convergence conditions of the multi-material AAPA program, determine the material fixed phase a and the activation term b, and execute the two-phase competition subroutine; (e) Determine the meshless EFGM global force load vector of the design domain based on the load magnitude applied in the brake disc design area; (f) Transform the design density field ρ to obtain the filtered density field and physical density field (g) Calculate the average density V of the physical density field within the scale control domain m and the approximate maximum density V max ; (h) Then calculate the scale control aggregation function g m ; (i) According to the given force and displacement boundary conditions, various force and displacement boundary conditions are processed, wherein the penalty function method is used to process the displacement boundary, and the meshless EFGM global force stiffness matrix penalty term and the meshless EFGM global force load vector penalty term are obtained; (j) According to the calculated physical density field and the meshless EFGM discrete control equations of the statics problem, combined with the meshless EFGM global force load vector determined in step (d) and the meshless EFGM global force stiffness matrix penalty term and meshless EFGM global force load vector penalty term obtained by the penalty function method in step (h), to solve the displacement parameter values ​​of the meshless EFGM nodes in the design domain; (k) Solve the displacement value of each meshless EFGM node by combining the displacement parameter value of the meshless EFGM node in the design domain; (l) Output the displacement parameter values, displacement values ​​and meshless EFGM global force load vector of the meshless EFGM nodes in the structural design domain based on the meshless EFGM; (m) Calculate the system flexibility C based on the displacement values ​​of the meshless EFGM nodes and the meshless EFGM overall force load vector; (n) Calculate the flexibility sensitivity difference dc between phases a and b diff 、dc diff Sensitivity dc to the initial design density field of the a-phase material, the difference between the volume of the a-phase material and the target volume V diff and volume sensitivity dv; (o) Convert the topological physical density field calculated by the current gridless EFGM into an RGB image as the input of the VGG-19 network model to extract image features; (p) Use the VGG-19 network model to extract the content and style features of the reference image; calculate the difference between the image features of the topological structure and the reference image in terms of content and style features, denoted as L style (T k ,R style ), L content (T k ,R style ), as the content loss term and style loss term of the loss function respectively; (q) Update network parameters through back-propagation; (r) Calculate physical density field Derivatives with respect to network parameters And calculate the style transfer network parameters Sensitivity to a-phase materials (s) Use the MMA algorithm to update the density of the a-phase material, the system flexibility C is used as the optimization target, and the a-phase material volume constraint V diff And the weighted style transfer loss function L diff =w l L diff,k (T k ,R) as optimization constraints; (t) Calculate the density of phase b material based on the total volume of phase a and phase b materials; (u) Input the iteration termination condition. The iteration termination condition is that the relative value of the volume fraction under the iteration step and the given volume constraint is less than 10 -3 And the relative values ​​of the objective function in the last ten iterations are all less than 10 -3 If the termination condition is met, the iteration terminates, and the optimal topology is output according to the physical density field of each meshless EFGM node. If not, steps (d)-(t) are repeated until the termination condition is met.

[0028] Figure 5-Figure 23 is the optimal topological structure obtained by the multi-material structure topology optimization based on the meshless EFGM and NST in this embodiment, where Figure 6 is the optimal topology of the multi-material structure when NST is not used in the embodiment of the present invention, Figure 7 is the optimal topological structure obtained when the style loss selection layer is Cov5_1 in the embodiment of the present invention, Figure 8 is the optimal topological structure obtained when the style loss selection layer is Cov45_1 in the embodiment of the present invention, Fig. 9 is the optimal topological structure obtained when the style loss selection layer is Cov345_1 in the embodiment of the present invention, Fig.10 is the optimal topological structure obtained when the style loss selection layer is Cov2345_1 in the embodiment of the present invention, Fig.11 is the optimal topological structure obtained when the style loss selection layer is Cov12345_1 in the embodiment of the present invention, Fig.12 is the optimal topological structure obtained when the content loss selection layer is Cov2_1 in the embodiment of the present invention, Fig.13 is the optimal topological structure obtained when the content loss selection layer is Cov3_1 in the embodiment of the present invention, Fig.14 is the optimal topological structure obtained when the content loss selection layer is Cov4_1 in the embodiment of the present invention, Fig.15 is the optimal topological structure obtained when the content loss selection layer is Cov5_1 in the embodiment of the present invention, Fig.16 is the optimal topological structure obtained when the content loss selection layer is Cov5_2 in the embodiment of the present invention, Fig.17 is the style loss weight coefficient w in the embodiment of the present invention s =1.0, content loss weight coefficient w c = 0, the optimal topology is obtained. Fig.18 is the style loss weight coefficient w in the embodiment of the present invention s =0.75, content loss weight coefficient w c =0.25, the optimal topology is obtained. Fig.19 is the style loss weight coefficient w in the embodiment of the present invention s =0.5, content loss weight coefficient w c = 0.5, the optimal topological structure is obtained. Fig. 20 is the style loss weight coefficient w in the embodiment of the present invention s =0.25, content loss weight coefficient w c =0.75, the optimal topology is obtained. Fig.21 is the style loss weight coefficient w in the embodiment of the present invention s =0, content loss weight coefficient w c =1.0, the optimal topology is obtained. Fig. 22 is the optimal topological structure when the minimum length scale control MinLSC is used instead of NST in the embodiment of the present invention, Fig.23 is the style loss weight coefficient w in the embodiment of the present invention s = 0.5 when the minimum length scale is used to control the optimal topology of MinLSC.

[0029] Although the present invention has been described in detail with reference to this embodiment, the above description does not limit the protection scope of the present invention, and any modifications and improvements based on the concept of the present invention are deemed to be within the protection scope of the present invention.

Claims

1. A multi-material structural topology optimization method based on meshless EFGM and style transfer, 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, material volume constraint η, material type q, meshless EFGM initial design density field ρ, topological structure content and style weight coefficient w c and w s , set the elastic modulus E1 and Poisson's ratio v of the material 12 , shear modulus G 12 , Poisson's ratio factor Bt, material orientation angle θ, given the design domain boundary conditions and load size, according to the meshless EFGM node information of the design domain, the background grid and the given boundary conditions, the Gaussian point information of the design domain is obtained; (2) Calculate the system stiffness matrix and load vector based on the meshless EFGM: (a) Determine the meshless EFGM overall force load vector of the design domain according to the magnitude of the force load applied in the design domain; (b) According to the given force and displacement boundary conditions, process various force and displacement boundary conditions, and use the penalty function method to process the displacement boundary to obtain the meshless EFGM overall force stiffness matrix penalty term and the meshless EFGM overall force load vector penalty term; (c) Calculate the material's elastic matrix based on the material's mechanical properties in, as well as are the elastic modulus and Poisson's ratio of the i-th material in the x and y directions in the material coordinate system, respectively, and satisfy the relationship The meshless EFGM global stiffness matrix of the design domain is assembled according to the meshless EFGM node relative density distribution function, and the meshless EFGM discrete control equations of the structural statics problem are established; (3) Length scale control of the design density field: (a) Filter the gridless EFGM design density field ρ to obtain the filtered density field In the formula v l represents the volume of the lth meshless EFGM node, and the weight coefficient ||x k -x l || represents the distance between the kth gridless EFGM node and the lth gridless EFGM node, d fil is the filter diameter; (b) the physical density field is then calculated based on the filter density field β represents the projection steepness, μ represents the threshold; (c) Calculate the average density of the physical density field in the scale control domain in w (k,l) =H(d min -2||x k -x l ||), d min represents the length scale controlling the diameter, H(·) represents the Heaviside function; (d) the sum of the weighted density is calculated based on the average density and the approximate maximum mean density Where p1 is the exponential factor; (e) Then calculate the scale control aggregation function in p1 is the exponential factor, N is the number of meshless EFGM nodes in the design domain, and ξ is the relaxation factor for scale control; (4) Calculate the difference between the topological structure obtained by the current gridless EFGM and the reference image based on the VGG-19 network framework: (a) Convert the physical density field based on the gridless EFGM The constructed topological structure is converted into an RGB image image1; (b) a reference image imgRef is defined for extracting the image style; (c) the VGG-19 network is used to extract the content and style features of the reference image imgRef; (d) the VGG-19 network is used to extract the pixel features of the image image1; (e) the difference between the pixel features of the image image1 and the extracted content and style pixel features of the reference image imgRef is calculated, which are the content loss L c and style loss L s ; (f) Multiply the content loss and style loss by the corresponding weight coefficient w c and w s Get the total loss of the network model Loss = w c L c +w s L s ; (g) Calculate the physical density field Gradients of the neural network parameters w and b And output parameters; (5) Calculate the density distribution of multi-material structures based on the alternating active-phase algorithm (AAPA): (a) Define the fixed phase a and the activated phase b of the material, where the value of the a-phase material is traversed in the range of [1, q], and the b-phase material is traversed in the sub-cycle, with a range of [a+1, q]; (b) According to the meshless EFGM global force load vector determined in step 3(a) and the meshless EFGM global force stiffness matrix penalty term and meshless EFGM global force load vector penalty term obtained by the penalty function method in step 3(b), solve the displacement parameter value of the meshless EFGM node in the design domain; solve the displacement value of each meshless node and the system flexibility C based on the displacement parameter value of the meshless EFGM node in the design domain; (c) Calculate the sensitivity C of the flexibility corresponding to the two materials a and b to the design variables a and C b , calculate the flexibility difference C1 = C a -C b And its filtering density field for a-phase material sensitivity dc, and volume sensitivity dv a (d) Calculation Filtering density field for a-phase material The derivative of (e) Calculate the difference V between the current volume of the a-phase material and the target volume diff_a , V diff_a As a volume constraint; the VGG-19 network output parameters Multiply by the weight factor w L Get the style constraint S diff_a (f) Update the physical density field of phase a material using the moving asymptotes algorithm (MMA) The system flexibility C is taken as the optimization target, and the volume constraint V diff_a and style constraint S diff_a As a constraint function; (g) Calculate the relative density ρ of the physical field of phase b material b ; (6) Based on the meshless EFGM and style transfer related theories, the mathematical model for multi-material structure topology optimization is established as follows: In the formula, represents the density of the jth material at the i-th meshless EFGM node, F is the meshless EFGM global force load column vector, and F α is the meshless EFGM total force load column vector penalty term, U is the meshless EFGM displacement column vector, To introduce the relative density value into the meshless EFGM global stiffness matrix, E0 is the elastic modulus of the solid material, E min In order to prevent the stiffness matrix from being singular, a smaller value of Young's modulus is given, is the global stiffness matrix penalty term of the meshless EFGM, H(·) is the Heaviside function, φ I is the meshless EFGM interpolation shape function, is the density of the jth material at the ith Gaussian point in the meshless EFGM, V represents the material volume, η (j) represents the volume fraction of the jth material, L diff,h is the loss function of the VGG-19 network model, δ is the relaxation factor used to judge the convergence of the loss function, and g m is the scale control constraint aggregation term, ξ is the relaxation factor for the scale control constraint, They represent the design density, filter density, and physical density of the j-th material at the i-th meshless EFGM node; (7) Define the iteration termination condition. If the termination condition is met, the iteration is terminated, and the optimal topological structure of multi-materials with scale control based on meshless EFGM and style transfer is output according to the physical field density value of each meshless EFGM node. If it is not met, continue to execute the subsequent steps and loop step (5) until the termination condition is met.

2. The multi-material structure topology optimization method based on meshless EFGM and style transfer according to claim 1, characterized in that Length scale control of the design density field: (a) Filtering the gridless EFGM design density field ρ to obtain the gridless EFGM filtered density field In the formula v l represents the volume of the lth meshless EFGM node, and the weight coefficient ||x k -x l || represents the distance between the kth gridless EFGM node and the lth gridless EFGM node, d fil is the filter diameter; (b) the physical density field is then calculated based on the filter density field Where β represents the projection steepness and μ represents the threshold; (c) Calculate the average density of the physical density field in the scale control domain in w (k,l) =H(d min -2||x k -x l ||), d min represents the length scale controlling the diameter, H(·) represents the Heaviside function; (d) the sum of the weighted density is calculated based on the average density and the approximate maximum mean density p1 is the exponential factor (e) and then the scale control aggregation function is calculated in p1 is the exponential factor, N is the number of meshfree EFGM nodes in the design domain, and ξ is the relaxation factor for scale control.

3. The multi-material structure topology optimization method based on meshless EFGM and style transfer according to claim 1, characterized in that The physical density field based on the meshless EFGM The constructed topological structure is converted into an RGB image image1; a reference image imgRef for extracting the image style is defined, and the VGG-19 network is used to extract the content and style features of the reference image imgRef.

4. The multi-material structure topology optimization method based on meshless EFGM and style transfer according to claim 1, characterized in that The VGG-19 network is used to extract the pixel features of image1, and the difference between the pixel features of image1 and the extracted content and style pixel features of the reference image imgRef is calculated, which are content loss L and style loss L respectively. c and style loss L s ; Multiply the content loss and style loss by the corresponding weight coefficient w c and w s Get the total loss of the network model Loss = w c L c +w s L s ; Calculate physical density field Gradients of the neural network parameters w and b And output the parameters.

5. The multi-material structure topology optimization method based on meshless EFGM and style transfer according to claim 1, characterized in that The density distribution of multi-material structures is calculated based on the alternating active-phase algorithm (AAPA): (a) Define the fixed phase a and the activated phase b of the material, where the value of the a-phase material is traversed in the range of [1, q], and the b-phase material is traversed in the sub-cycle, and the range of change is [a+1, q]; (b) According to the meshless EFGM global force load vector determined in step 2(a) and the meshless EFGM global force stiffness matrix penalty term and meshless EFGM global force load vector penalty term obtained by the penalty function method in step 2(b), the displacement parameter values ​​of the meshless EFGM nodes in the design domain are solved; Based on the displacement parameter values ​​of the meshless EFGM nodes in the design domain, the displacement values ​​of each meshless node and the system flexibility C are solved; (c) Calculate the sensitivity C of the flexibility corresponding to the two materials a and b to the design variables a and C b , calculate the flexibility difference C1 = C a -C b And its filtering density field for a-phase material sensitivity dc, and volume sensitivity dv a (d) Calculation Filtering density field for a-phase material The derivative of (e) Calculate the difference V between the current volume of the a-phase material and the target volume diff_a , V diff_a As a volume constraint; the VGG-19 network output parameters Multiply by the weight factor w L Get the style constraint S diff_a (f) Update the physical density field of phase a material using the moving asymptotes algorithm (MMA) The system flexibility C is taken as the optimization target, and the volume constraint V diff_a and style constraint S diff_a As a constraint function; (g) Calculate the relative density of the physical field of phase b material

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