Topological optimization method of neural network

Abstract

The invention relates to a topological optimization method of a neural network. According to the method, density structure design is carried out by constructing a feedforward neural network, and intelligent optimization design of structure topology is realized by combining finite element analysis, sensitivity analysis and a gradient optimization algorithm. According to the method, space coordinates are mapped into unit density values through the nonlinear mapping capacity of a neural network, the gradient is calculated through a back propagation algorithm, network parameters are updated through a gradient descent method, and automation and optimization of structural design are achieved while physical constraints are kept.

Description

Technical Field

[0001] This invention relates to structural design and optimization methods, and particularly to a topology optimization method based on neural networks. Background Technology

[0002] Traditional topology optimization methods have played a crucial role in engineering design, but they also suffer from numerous limitations. These methods heavily rely on finite element analysis and complex numerical calculations, consuming significant computational resources and time. For example, when dealing with large-scale structural optimization problems, the computational complexity increases dramatically, leading to lengthy optimization processes. Furthermore, traditional methods typically require numerous design variables and iterative processes; the discretization of the design space limits design flexibility and innovation, making it difficult to meet the demands of modern engineering for efficient and high-precision design. With the advancement of technology, the rise of deep learning has brought new opportunities to topology optimization. Neural networks possess powerful mapping capabilities and can learn complex data patterns, potentially overcoming the shortcomings of traditional methods. However, applying deep learning to topology optimization faces challenges, such as effectively embedding physical constraints, handling high-dimensional design spaces, and improving optimization efficiency and accuracy. This invention aims to leverage the advantages of neural networks to solve the problems of traditional topology optimization methods, achieving more efficient and flexible structural design optimization. Summary of the Invention

[0003] Purpose of the invention: The purpose of this invention is to provide a topology optimization method based on neural networks, specifically a topology optimization method based on physical neural network embedding.

[0004] Technical solution: The aforementioned neural network-based topology optimization method includes the following steps:

[0005] S1: Construct a neural network architecture, using a feedforward neural network, and determine the number of network layers, the number of neurons in each layer, and the type of activation function. The neural network is used to establish a nonlinear mapping relationship between the design spatial coordinates and the material density distribution.

[0006] S2: Input design space coordinates, take the center coordinates of each unit in the design domain as the input of the neural network, the coordinate information includes (x, y) coordinates in two-dimensional case or (x, y, z) coordinates in three-dimensional case;

[0007] S3: Output density value, which is calculated by forward propagation of a neural network to obtain the material relative density value corresponding to the input coordinate position. The density value ranges from 0 to 1, where 0 represents voids and 1 represents solid material.

[0008] S4: Finite element analysis is used to calculate the structural response. Based on the density values ​​output by the neural network, the material properties of each element are calculated using the SIMP interpolation method. The global stiffness matrix is ​​assembled, the static equilibrium equations are solved to obtain the displacement field, and the flexibility of the structure is calculated.

[0009] S5: Calculate the sensitivity value, and calculate the loss function value based on the objective function. The loss function includes compliance loss and volume loss.

[0010] S6: Backpropagation adjusts the network weights. Using automatic differentiation and the chain rule, the gradient of the objective function with respect to the neural network weights and biases is calculated. The network parameters are updated using gradient descent or its improved algorithm.

[0011] S7: Check the convergence conditions and determine whether the termination conditions are met, including the change in the objective function value between two adjacent iterations being less than a preset threshold, reaching the maximum number of iterations, or the proportion of gray units being less than a set value.

[0012] S8: Iterate and update or output the results. If the convergence condition is not met, return to step S2 to continue iterative optimization. If the convergence condition is met, output the final density field distribution and topology.

[0013] Specifically,

[0014] S1: Construct a bidirectional dynamic architecture of displacement network and density network. A fully connected feedforward neural network is adopted, using a sinusoidal representation network (SIREN) as the activation function. Its forward propagation formula is:

[0015] h j =α(W j h j -1+b j ), j=1,2,..., Lρ=σ(W L h L -1+b L )

[0016] Among them, h j For the hidden state of the j-th layer, W j and b j Here, α represents the weights and biases, σ represents the activation function of the hidden layer, σ represents the activation function of the output layer, and L represents the number of network layers.

[0017] S2: Displacement network design, with spatial coordinates as input and displacement components as output.

[0018]

[0019] Where xnorm is the normalized coordinate, X is the coordinate matrix, N is the total number of cells, and d is the spatial dimension.

[0020] S3: The output density value is calculated through forward propagation of a neural network, embedding physical constraints and dynamic parameter updates. External force boundary conditions are directly applied to the external force work term in the loss function. The Adam optimizer is used to jointly train the displacement network parameters θ and the density network parameters ρ.

[0021] ρ=f NN (X; θ)=[ρ1, ρ2,…,ρ N ] T ρ i ∈[0,1],i=1,2,…,N

[0022] Among them, fN N Let θ be the neural network mapping function, θ be the network parameter vector, and ρ be the network parameter vector. i Let be the density value of the i-th cell.

[0023] S4: Finite element analysis calculates the structural response. Based on the density values ​​output by the neural network, the material properties of each element are calculated using the SIMP interpolation method, and the density distribution is updated. The density network ensures that the volume constraint is explicitly considered in its loss function. The static equilibrium equations are solved to obtain the displacement field, and the structural flexibility, stress and other response quantities are calculated.

[0024]

[0025] Among them, E i Let K be the Young's modulus of the i-th element, and K be the global stiffness matrix. Let U be the element stiffness matrix, U be the displacement vector, and C be the structural flexibility.

[0026] S5: Calculate the loss function value and sensitivity value based on the objective function. The loss function includes compliance loss and volume loss; the first part is the compliance loss, which can be expressed as:

[0027]

[0028] Where C0 is the initial compliance of the structure, using this term as a divisor can regularize the compliance loss, preventing it from becoming too large. The second part, volumetric loss, can be expressed as:

[0029]

[0030] Therefore, the total loss can be expressed as:

[0031]

[0032] Where λ is an adaptive loss weight term, used to dynamically program the impact of flexibility loss and volume loss on the total loss. This value is related to the number of iterations, and the calculation formula is as follows:

[0033] λ = min{λ0 + epoch * Δλ, λ max}

[0034] Where λ0 is a small initial weight value, epoch is the current iteration number, and λ maxΔλ represents the maximum weight value, and Δλ represents the weight increment.

[0035] S6: Backpropagation adjusts network weights, and Gaussian integration is used to calculate strain energy. Element strain energy is calculated through Gaussian integration points, decoupling the material mapping relationship.

[0036]

[0037] Where, m t and v t These are the first-order and second-order moment estimates, respectively, where α is the learning rate and β1 and β2 are momentum parameters.

[0038] Where λ and u are Lamé constants;

[0039] S7: Check the convergence conditions and determine whether the termination conditions are met, including the change in the objective function value between two adjacent iterations being less than a preset threshold, reaching the maximum number of iterations, or the proportion of grayscale units being less than a set value. Then, perform iterations. The iteration termination condition is:

[0040]

[0041] The first condition is that the change in the objective function is less than a threshold ε, and the second condition is that the maximum number of iterations k is reached. max The third condition is that the proportion of grayscale units is less than the threshold τ, and the grayscale unit is defined as:

[0042] N gray =#i:0.1<ρ i <0.9.

[0043] S8: Output the result or continue iteration. If the convergence condition is not met, return to the coordinate input step to continue the optimization loop; if convergence has been achieved, output the optimal density field distribution and the final topology. The interpolation formula for the element material properties in the SIMP method is:

[0044]

[0045] Where E0 represents the Young's modulus of the solid material, E min It is the minimum value assigned to an empty cell, p is the penalty factor, and ρ is the minimum value. i It is the density value output by the neural network.

[0046] Compared with the prior art, the present invention has the following advantages:

[0047] This invention significantly improves the efficiency and accuracy of traditional topology optimization by introducing two cooperating neural networks—a displacement network and a density network. By directly embedding physical constraints into the neural networks, this method strictly adheres to physical laws during the optimization process, while greatly reducing reliance on large-scale datasets and sensitivity analysis. This innovative optimization strategy not only improves computational efficiency but also explores better solutions in a wider design space, thereby meeting the higher requirements of modern engineering design for material strength, stiffness, durability, and reliability.

[0048] This invention provides a method for finding the optimal topology for material distribution under given loads and constraints, thereby improving structural performance and reducing material costs. This invention is primarily applied in industries such as machinery, aerospace, automotive manufacturing, and construction, which require structural design and optimization. It achieves innovative optimization of structural topology through neural network technology. Attached Figure Description

[0049] Figure 1 This is a flowchart of the topology optimization method for a neural network in this invention;

[0050] Figure 2 This is a flowchart of the displacement and density physical network in this invention;

[0051] Figure 3 This is a schematic diagram of an example from the present invention;

[0052] Figure 4 This is a diagram showing the topology optimization results of an example of the present invention;

[0053] Figure 5 This is a comparison chart of the optimization results of this invention and the SIMP method. Detailed Implementation

[0054] To make the objectives, technical solutions, and advantages of the present invention clearer, the technical solutions of the present invention will be further described below.

[0055] Combination Figure 1 The present invention adopts the following technical solution and is implemented in accordance with the eight steps shown in the flowchart:

[0056] Taking the topology optimization of a cantilever beam as an example, the design domain is a rectangular region of 100×50, the left end is fixed, and a downward load F=1000N is applied to the midpoint of the right end, with a volume fraction constraint of 0.5.

[0057] Step S1: Construct the neural network architecture;

[0058] A feedforward neural network is used, with the following specific parameter settings:

[0059] Input layer: 2 neurons (corresponding to x and y coordinates)

[0060] Hidden layers: 4 layers, 128 neurons per layer

[0061] Output layer: 1 neuron (output density value)

[0062] Hidden layer activation function: ReLU function, α(x) = max(0, x)

[0063] Output layer activation function: Sigmoid function.

[0064] The formula for network forward propagation is:

[0065] h j =α(Wjhj-1+b j ), j=1,2,3,4ρ=σ(W5h4+b5)

[0066] The network parameters are initialized using the Xavier initialization method, with the weight matrix initialized randomly and the bias initialized to zero.

[0067] Step S2: Input the design space coordinates;

[0068] The 100×50 design domain was meshed, resulting in 5000 quadrilateral elements (100×50 grid). The center coordinates (x, y, z) of each element were extracted. i y i ), where i = 1, 2, ..., 5000.

[0069] Normalize the coordinates:

[0070]

[0071] Construct the input coordinate matrix:

[0072]

[0073] Step S3: Output density value;

[0074] Normalized coordinates are input into the neural network, and the density values ​​of each unit are calculated through forward propagation.

[0075] ρ=f NN (X; θ)=[ρ1, ρ2,…,ρ 5000 ] T

[0076] Each density value ρ i ∈[0, 1], where 0 represents void material and 1 represents solid material. In the initial iteration, all density values ​​are typically close to 0.5 (uniform distribution).

[0077] Step S4: Calculate the structural response using finite element analysis;

[0078] Based on the density values ​​output by the neural network, the Young's modulus of each unit is calculated using the SIMP interpolation method:

[0079]

[0080] Where, E0 = 210000MPa (Young's modulus of steel), E min =0.001MPa, penalty factor p=3, Poisson's ratio ν=0.3.

[0081] Assemble the global stiffness matrix:

[0082] in The stiffness matrix is ​​for a standard quadrilateral element (8×8).

[0083] Apply boundary conditions (left end fixed) and load conditions (midpoint load on the right end), and solve the linear equation system:

[0084] K(ρ)U=F

[0085] Calculate structural flexibility:

[0086] C(ρ)=F T U = U T K(ρ)U

[0087] Step S5: Calculate the sensitivity value;

[0088] A loss function is constructed based on the objective function. The loss function consists of two parts: flexibility loss and volume loss. Adaptive weighting coefficients are used to balance the influence of the two on the total loss. The specific calculation steps are as follows:

[0089] Structural flexibility is the primary optimization objective; lower flexibility indicates better structural stiffness performance. The flexibility loss function is defined as:

[0090]

[0091] Where C0 is the compliance of the initial structure, using this term as a divisor can regularize the compliance loss and avoid compliance issues.

[0092] The volume loss value is too large. To satisfy the volume fraction constraint, the volume loss function is defined as follows:

[0093]

[0094] Combining flexibility loss and volume loss, the total loss function is defined as:

[0095]

[0096] Where λ is an adaptive weighting coefficient used to dynamically adjust the relative importance of compliance loss and volume loss. This coefficient is dynamically updated during the iteration process, and its calculation formula is:

[0097] λ = min{λ0 + epoch * Δλ, λ max}

[0098] Where λ0 = 0.01 is the initial weight value, epoch is the current iteration number, Δλ = 0.001 is the weight growth rate, and λ max =0.1 is the maximum weight value. By adaptively adjusting the weights, flexibility is reduced in the early stages of optimization, and volume constraint satisfaction is strengthened in the later stages.

[0099] Step S6: Backpropagation to adjust network weights;

[0100] Calculate the gradient of the objective function with respect to the network parameters using the chain rule:

[0101]

[0102] in It was calculated using automatic differentiation techniques.

[0103] The Adam optimization algorithm is used to update network parameters, and hyperparameters are set as follows:

[0104] Learning rate: α = 0.001

[0105] First-order moment attenuation coefficient: β1 = 0.9

[0106] Second-order moment attenuation coefficient: β² = 0.999

[0107] Numerical stability term: ∈ = 1 × 10 -8

[0108] Updated formula:

[0109] Step S7: Check the convergence conditions;

[0110] Set convergence criteria:

[0111] Threshold for change in objective function: ε = 1 × 10 -6

[0112] Maximum number of iterations: k max =1000

[0113] Grayscale unit ratio threshold: T = 0.05

[0114] Check after each iteration:

[0115] The number of grayscale units: N gray =#i: 0.1 < ρi < 0.9.

[0116] At the 268th iteration, the objective function changed by 8.5 × 10⁻⁶. -7 <1×10 -6 The convergence condition is met.

[0117] Step S8: Iterate and update or output the results.

[0118] The iteration process stops because the convergence condition is met on the 268th iteration.

[0119] The final output result is:

[0120] Density field distribution matrix: ρ final = [0.9821, 0.0034, 0.8765, ..., 0.0012] T

[0121] Final topology: After binarization with a threshold of 0.5, a clear black and white topology diagram is obtained, showing the optimal topology of a typical cantilever beam.

[0122] The performance parameters are as follows:

[0123] Final compliance value: C final =2.8432×10 -3

[0124] Actual volume fraction: V f，final =0.4998

[0125] Number of iterations: 268

[0126] Optimized time: 45.3 seconds

[0127] Grayscale unit ratio: 0.032

[0128] If the convergence condition is not met, return to step S2, recalculate the coordinate mapping and density distribution using the updated network parameters, and continue the next round of iterative optimization.

[0129] Through the complete implementation of the above eight steps, the topology optimization of the cantilever beam structure was successfully achieved, verifying the effectiveness of the method of the present invention.

[0130] Application Example 1

[0131] This invention provides a specific implementation example of a topology optimization method based on neural networks. The initial learning rate is 1e-3, and the Young's modulus E = 210 GPa. A schematic diagram of the example is shown below. Figure 3 As shown.

[0132] Case 1) Figure 3 As shown, the volume fraction of the object, v f =0.5, and the left side is fixed. A force F is applied to the midpoint of its right side, and topology optimization is performed. Using the neural network-based topology optimization method of this invention, the topology optimization result is as follows. Figure 3 As shown, its flexibility value is 2.8432.

[0133] Case 2) Figure 3 As shown, the volume fraction ν of the object f =0.7, and the left side is fixed. A force F is applied to the bottom right side, and topology optimization is performed on it. Using the neural network-based topology optimization method of this invention, the topology optimization result is as follows. Figure 3 As shown, its compliance value is 1.2491.

[0134] Case 3) Figure 3 As shown, the volume fraction ν of the object f =0.45, and with the left and right sides fixed, an average force F is applied above it. Topology optimization is performed on it, and the topology optimization method based on neural networks of this invention is used to obtain the topology optimization result as follows. Figure 3 As shown, its flexibility value is 374.5596.

[0135] Case 4) Figure 3 As shown, the volume fraction of the object, v f =0.3, and the bottom two sides are fixed. A force F is applied downward at the midpoint below it. Topology optimization is performed on it. Using the neural network-based topology optimization method of this invention, the topology optimization result is as follows. Figure 4 As shown, its flexibility value is 1.4582.

[0136] As can be seen from this implementation case, the neural network-based topology optimization method of the present invention has greatly improved topology optimization, and can better optimize the structure of materials. It can not only improve the efficiency of the optimization process, but also explore a wider design space while maintaining physical constraints, thereby achieving a better design solution. It brings an innovative method to the field of topology optimization to meet the increasing complexity and performance requirements in modern engineering design.

[0137] Furthermore, a comparison between the optimization of the case using the neural network-based topology optimization method of this invention and the conventional optimization of the case using the SIMP method shows that the method yields a smaller compliance value and better optimization results.

[0138] Case 1 Case 2 Case 3 Case 4 The compliance value of this topology optimization method 2.8432 1.2491 374.5596 1.4582 Compliance value of SIMP method 5.0871 1.9994 406.4573 1.7437

[0139] The results show that the neural network-based topology optimization method of this invention outperforms the SIMP method in terms of compliance value, significantly improving optimization efficiency and quality.

[0140] This invention establishes a nonlinear mapping relationship between spatial coordinates and material density distribution by constructing a neural network, providing a highly efficient topology optimization method. This method can better explore the design space and meet the higher demands of modern engineering design for complexity and performance. This approach not only improves optimization efficiency but also achieves a superior design solution while maintaining physical constraints.

[0141] The above are merely preferred embodiments of the present invention and do not constitute any limitation on the present invention. Any equivalent substitutions or modifications made by those skilled in the art to the technical solutions and content disclosed in the present invention without departing from the scope of the present invention shall be deemed to have remained within the protection scope of the present invention.

Claims

1. A topology optimization method based on neural networks, characterized in that, Includes the following steps: S1: Construct a neural network architecture, using a feedforward neural network, and determine the number of network layers, the number of neurons in each layer, and the type of activation function. The neural network is used to establish a nonlinear mapping relationship between the design spatial coordinates and the material density distribution. S2: Input design space coordinates, take the center coordinates of each unit in the design domain as the input of the neural network, the coordinate information includes (x, y) coordinates in two-dimensional case or (x, y, z) coordinates in three-dimensional case; S3: Output density value, which is calculated by forward propagation of a neural network to obtain the material relative density value corresponding to the input coordinate position. The density value ranges from 0 to 1, where 0 represents voids and 1 represents solid material. S4: Finite element analysis is used to calculate the structural response. Based on the density values ​​output by the neural network, the material properties of each element are calculated using the SIMP interpolation method. The global stiffness matrix is ​​assembled, the static equilibrium equations are solved to obtain the displacement field, and the flexibility of the structure is calculated. S5: Calculate the sensitivity value, and calculate the loss function value based on the objective function. The loss function includes compliance loss and volume loss. S6: Backpropagation adjusts the network weights. Using automatic differentiation and the chain rule, the gradient of the objective function with respect to the neural network weights and biases is calculated. The network parameters are updated using gradient descent or its improved algorithm. S7: Check the convergence conditions and determine whether the termination conditions are met, including the change in the objective function value between two adjacent iterations being less than a preset threshold, reaching the maximum number of iterations, or the proportion of gray units being less than a set value. S8: Iterate and update or output the results. If the convergence condition is not met, return to step S2 to continue iterative optimization. If the convergence condition is met, output the final density field distribution and topology.

2. The topology optimization method based on neural networks according to claim 1, characterized in that, The construction of the neural network architecture specifically includes: designing an input layer to receive spatial coordinates, setting multiple hidden layers for nonlinear mapping, and an output layer to generate density values; the forward propagation formula of the neural network is: h j =α(W j h j -1+b j ),j=1,2,…,Lρ=σ(W L h L -1+b L ) Among them, h j For the hidden state of the j-th layer, W j and b j Here, α represents the weights and biases, σ represents the activation function of the hidden layer, σ represents the activation function of the output layer, and L represents the number of network layers.

3. The topology optimization method based on neural networks according to claim 1, characterized in that, The process of inputting the design space coordinates includes: dividing the design domain into grids, extracting the center coordinates of each grid cell, and normalizing the coordinates to use them as the input vector of the neural network. The specific formula is as follows: Where, x norm Let X be the normalized coordinates, N be the coordinate matrix, d be the total number of cells, and d be the spatial dimension.

4. The topology optimization method based on neural networks according to claim 1, characterized in that, The output density value is achieved as follows: the output of the last layer of the neural network is processed by an activation function to obtain a continuous value between 0 and 1, which represents the relative density of the material at the corresponding location. p=f NN (X;θ)=[ρ1,ρ2,…,ρ N ] T r i ∈[0,1],i=1,2,…,N Among them, f NN Let θ be the neural network mapping function, θ be the network parameter vector, and ρ be the network parameter vector. i Let be the density value of the i-th cell.

5. The topology optimization method based on neural networks according to claim 1, characterized in that, The finite element analysis calculation of structural response includes: calculating the element stiffness coefficients based on the density values ​​using the SIMP method, assembling the global stiffness matrix according to finite element theory, and solving for the displacements. The specific formula is as follows: Among them, E i Let K be the Young's modulus of the i-th element, and K be the global stiffness matrix. Let U be the element stiffness matrix, U be the displacement vector, and C be the structural flexibility.

6. The topology optimization method based on neural networks according to claim 1, characterized in that, In the optimization described above, compliance is the objective function. The smaller the compliance, the better the structure is considered to be. Volume constraints also need to be considered. Therefore, the loss function consists of two parts: the first part is the compliance loss, which can be expressed as: Where C0 is the compliance of the initial structure, using this term as a divisor can regularize the compliance loss and prevent it from becoming too large. The second part, the volume loss, can be expressed as: Therefore, the total loss can be expressed as: Where λ is an adaptive loss weight term, used to dynamically program the impact of flexibility loss and volume loss on the total loss. This value is related to the number of iterations, and the calculation formula is as follows: λ=min{λ0+epoch*Δλ,λ max } Where λ0 is a small initial weight value, epoch is the current iteration number, and λ max Δλ represents the maximum weight value, and Δλ represents the weight increment.

7. The topology optimization method based on neural networks according to claim 1, characterized in that, The backpropagation adjustment of network weights is achieved through the following steps: calculating the gradient of the objective function with respect to the network weights using the chain rule, and updating the network parameters using the Adam optimization algorithm: Where, m t and v t These are the first-order and second-order moment estimates, respectively, where α is the learning rate and β1 and β2 are momentum parameters.

8. The topology optimization method based on neural networks according to claim 1, characterized in that, The convergence criteria for the check include one or a combination of the following judgment criteria: The first condition is that the change in the objective function is less than a threshold ε, and the second condition is that the maximum number of iterations k is reached. max The third condition is that the proportion of grayscale units is less than the threshold τ, and the grayscale unit is defined as: N gray =#i:0.1<ρ i <0.9 9. The topology optimization method based on neural networks according to claim 1, characterized in that, During the iterative update process, if the convergence condition is not met, the current network parameters are maintained, and the design space coordinates are re-entered for the next forward propagation and optimization iteration; if the convergence condition is met, the iteration is stopped and the optimized density field distribution is output.

10. The topology optimization method based on neural networks according to claim 1, characterized in that, The interpolation formula for the element material properties in the SIMP method is as follows: Where E0 represents the Young's modulus of the solid material, E min It is the minimum value assigned to the empty cell, p is the penalty factor, and ρ is the minimum value. i It is the density value output by the neural network.

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