Multi-scale structure rigidity optimization design method based on neural network re-parameterization

Abstract

The invention discloses a multi-scale structural stiffness optimization design method based on neural network re-parameterization, which comprises the following steps: microstructure parameterization modeling: carrying out parameterization modeling on a microstructure by adopting a level set method, and controlling the volume fraction and geometric configuration of the microstructure by adjusting a cutting height parameter; proxy model construction: establishing a mapping relation between cutting height parameters and microstructure equivalent mechanical properties by using a neural network to form a high-precision proxy model; neural network re-parameterization design: constructing a neural network optimizer, taking coordinates of each discrete unit in a design domain as input, and outputting corresponding cutting height parameters; neural network parameters are updated through automatic differentiation and back propagation, and continuous optimization of design variables is achieved; and a multi-scale topological optimization process: calculating a global stiffness matrix and structural flexibility in combination with an agent model and finite element analysis, and obtaining an optimal multi-scale structural topology meeting volume constraints through loss function minimization iterative optimization.

Description

Technical Field

[0001] The present invention relates to the technical field of structural optimization design, and particularly relates to a multi-scale structural stiffness optimization design method based on neural network reparameterization. Background Art

[0002] With the increasing complexity of structural design requirements, topology optimization (TO), as an efficient design method, has been widely applied in fields such as aerospace and automotive manufacturing. Topology optimization maximizes structural performance (such as stiffness and lightweight) by optimizing the distribution of materials within a given design domain. After years of development, topology optimization methods have derived various classic techniques, including the solid isotropic material with penalization (SIMP), level set method (LSM), evolutionary structural optimization (ESO), etc., and have achieved remarkable results in single-scale structural optimization.

[0003] In recent years, with the breakthrough of additive manufacturing technology, the design and manufacturing of multi-scale structures have become possible. Multiscale topology optimization (MTO) can significantly improve the mechanical properties and functional diversity of structures by synergistically optimizing the material distribution at the macro and micro scales. Existing MTO methods are mainly divided into two categories: Uniform microstructure optimization: The same type of microstructure (such as periodic lattice) is adopted in the macrostructure, and the microconfiguration is controlled by a small number of parameters. Although this method reduces the computational complexity, it limits the design freedom and is difficult to achieve performance breakthroughs. Non-uniform microstructure optimization: Differentiated microstructures are adopted in different regions to maximize local performance. However, this method faces two major challenges. One is the high computational cost: nested execution of microscopic homogenization analysis and macroscopic finite element calculation leads to huge consumption of computational resources. The other is poor microstructure connectivity: there is a lack of geometric constraints between non-uniform microstructures, which easily leads to interface discontinuity problems, affecting the load-bearing capacity of the structure and increasing the process difficulty of additive manufacturing. Summary of the Invention

[0004] To solve some or all of the above technical problems existing in the prior art, the present invention provides a multi-scale structural stiffness optimization design method based on neural network reparameterization.

[0005] The technical solution of the present invention is as follows:

[0006] A multi-scale structural stiffness optimization design method based on neural network reparameterization is provided, and the method includes:

[0007] Microstructure parametric modeling: The level set method is used to perform parametric modeling on the microstructure, and the volume fraction and geometric configuration of the microstructure are controlled by adjusting the cutting height parameter;

[0008] Surrogate model construction: Use a neural network to establish a mapping relationship between the cutting height parameter and the equivalent mechanical properties of the microstructure, forming a high-precision surrogate model for quickly predicting the elastic matrix and volume fraction of the microstructure;

[0009] Neural network reparameterization design: Construct a neural network optimizer that takes the coordinates of each discrete unit in the design domain as input and outputs the corresponding cutting height parameter; Update the neural network parameters through automatic differentiation and backpropagation to achieve continuous optimization of the design variables;

[0010] Multi-scale topology optimization process: Combine the surrogate model and finite element analysis to calculate the global stiffness matrix and structural compliance, and iteratively optimize through minimizing the loss function to obtain the optimal multi-scale structural topology that meets the volume constraint.

[0011] In an embodiment of the present invention, the level set method generates microstructures with different volume fractions by fixing the level set function and adjusting the height of the cutting plane, and the range of its volume fraction is from 0.2 to 0.8.

[0012] In an embodiment of the present invention, the steps of constructing the surrogate model include dataset generation: Uniformly sample the cutting height parameter through the homogenization theory to generate data of the elastic matrix and volume fraction of the microstructure for training the surrogate model.

[0013] In an embodiment of the present invention, in the step of dataset generation, the sampling interval of the cutting height parameter is uniformly equally spaced, and the division ratio of the training set to the validation set is 9:1.

[0014] In an embodiment of the present invention, the surrogate model uses a fully connected neural network, whose input is the cutting height parameter, and the output is the equivalent elastic matrix and volume fraction of the microstructure, and is trained with the mean absolute error as the loss function.

[0015] In an embodiment of the present invention, the output layer of the neural network optimizer uses the Sigmoid activation function to limit the cutting height parameter within a preset range; The hidden layer uses the LeakyReLU activation function to enhance the non-linear expression ability.

[0016] In an embodiment of the present invention, in the multi-scale topology optimization process, the loss function is defined as a weighted combination of the structural compliance and the volume fraction constraint, and the Adam optimizer automatically calculates the gradient and updates the neural network parameters.

[0017] In an embodiment of the present invention, in the multi-scale topology optimization process, the global stiffness matrix is obtained by combining the element stiffness matrices output by the surrogate model and is used to solve the displacement vector and the structural compliance.

[0018] In one embodiment of the present invention, the method is applicable to additive manufacturing technology, and the optimized multi-scale structure has non-uniform micro-space distribution and continuous connectivity.

[0019] The main advantages of the technical solution of the present invention are as follows:

[0020] The multi-scale structure stiffness optimization design method based on neural network reparameterization of the present invention encodes the design variables (cutting height parameters) in multi-scale topology optimization as network weights through neural network reparameterization technology, and uses automatic differentiation to achieve efficient calculation of gradients, avoiding complex manual derivation in traditional sensitivity analysis. The introduction of the surrogate model significantly reduces the computational cost of homogenization analysis, enabling the optimization process to be completed with ordinary computing resources. The finally designed multi-scale structure has both high stiffness and lightweight characteristics, and is suitable for the design of complex components in fields such as aerospace and automotive. Description of the Drawings

[0021] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention, and those of ordinary skill in the art can obtain other drawings based on these drawings without creative efforts.

[0022] Figure 1 It is a flowchart of the multi-scale structure stiffness optimization design method based on neural network reparameterization according to an embodiment of the present invention;

[0023] Figure 2 It is a schematic diagram of the microstructure generated by the level set method in the multi-scale structure stiffness optimization design method based on neural network reparameterization according to an embodiment of the present invention;

[0024] Figure 3 It is a schematic diagram of different microstructures in the multi-scale structure stiffness optimization design method based on neural network reparameterization according to an embodiment of the present invention;

[0025] Figure 4 It is a neural network framework diagram in the multi-scale structure stiffness optimization design method based on neural network reparameterization according to an embodiment of the present invention;

[0026] Figure 5 It is a flowchart of the neural network reparameterization multi-scale structure optimization in the multi-scale structure stiffness optimization design method based on neural network reparameterization according to an embodiment of the present invention. Detailed Embodiments

[0027] To make the objectives, technical solutions, and advantages of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below in conjunction with specific embodiments of the present invention and the corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of the present invention.

[0028] The following will detail the technical solutions provided by the embodiments of the present invention in conjunction with the drawings.

[0029] The embodiments of the present invention provide a multi-scale structural stiffness optimization design method based on neural network reparameterization. As shown in the attached Figure 1 figures, it includes:

[0030] S1, Microstructure parametric modeling: Parametrically model the microstructure using the level set method, and control the volume fraction and geometric configuration of the microstructure by adjusting the cutting height parameter.

[0031] The purpose of this step is to parametrically model the microstructure, control the volume fraction and geometric configuration of the microstructure by adjusting the cutting height, and provide a basis for establishing the dataset in the subsequent steps.

[0032] Among them, the level set method implicitly describes the boundary of the microstructure through a high-dimensional function. For example, select a preset level set function (such as a hyperbolic function), and its zero isosurface represents the surface of the microstructure. By adjusting the height parameter of the cutting plane, the geometric shape of the microstructure can be dynamically changed.

[0033] The cutting height parameter determines the position of the cutting plane in the level set function. For example: when the cutting height is set to a lower value, the cutting plane is located at the bottom of the level set function, generating a microstructure with a higher volume fraction (close to fully filled material). When the cutting height gradually increases, the cutting plane moves upward, and the volume fraction of the microstructure decreases accordingly, forming more holes or sparse structures. By continuously adjusting the cutting height parameter, microstructures with different volume fractions can be generated within the design domain to ensure the balance between lightweight and mechanical properties of the structure.

[0034] The geometric configuration of each microstructure is uniquely determined by the cutting height parameter. For example, use the cutting height parameter as the input of the subsequent surrogate model to predict the equivalent mechanical properties of the microstructure.

[0035] S2, Surrogate model construction: Use a neural network to establish a mapping relationship between the cutting height parameter and the equivalent mechanical properties of the microstructure, forming a high-precision surrogate model for quickly predicting the elastic matrix and volume fraction of the microstructure.

[0036] This step constructs a surrogate model to quickly predict the equivalent elastic matrix and volume fraction of the microstructure, avoiding the high computational cost of traditional homogenization analysis. The specific construction process is as follows:

[0037] Data generation and sampling: Within the design range of the cutting height parameter (e.g., from 0 to 1), uniform equidistant sampling is performed to generate multiple sample points. For each sample point, the equivalent elastic matrix components (such as elastic modulus, Poisson's ratio) and volume fraction of the microstructure are calculated through numerical homogenization theory to form a complete dataset.

[0038] Neural network structure and training: The input of the surrogate model is a single cutting height parameter, and the output is the equivalent elastic matrix components and volume fraction corresponding to this parameter. A fully connected neural network is adopted, which includes multiple hidden layers (e.g., 3 layers), with a certain number of neurons (e.g., 64) set in each layer, and the ReLU activation function is used to enhance the non-linear expression ability. By minimizing the mean absolute error (MAE) between the predicted value and the true value, the network weights are optimized to ensure the high accuracy of the surrogate model.

[0039] Model validation and deployment: The dataset is divided into a training set and a validation set to verify the generalization ability of the surrogate model. After training, the surrogate model can directly replace the traditional homogenization calculation, shortening the solution time of the mechanical properties of the microstructure from hours to milliseconds.

[0040] S3, Neural network reparameterization design: A neural network optimizer is constructed, taking the coordinates of each discrete unit in the design domain as the input and outputting the corresponding cutting height parameter; the neural network parameters are updated through automatic differentiation and backpropagation to achieve continuous optimization of the design variables.

[0041] This step aims to encode the design variables (cutting height) as network parameters through a neural network to achieve automated optimization.

[0042] The neural network architecture is designed as follows: The input layer receives the coordinate information of each discrete unit in the design domain (such as two-dimensional coordinates x and y). The hidden layer adopts a multi-layer fully connected structure (e.g., 4 layers, with 128 neurons in each layer), and the non-linear mapping ability is enhanced through activation functions. The output layer outputs the initial cutting height parameter corresponding to each unit, and the parameter is constrained within a preset range (e.g., from 0 to 1) through the Sigmoid function.

[0043] The parameter update and optimization mechanism includes: In the forward propagation process, the design domain coordinates are input into the network to obtain the initial cutting height parameter. In the backpropagation process, the gradient of the loss function with respect to the network weights is calculated through automatic differentiation, and the Adam optimizer is used to update the network parameters. In the physical constraint conversion process, the initial cutting height parameter output by the network is linearly mapped to the actual design range (for example, from 0.2 to 0.8) to ensure that the generated microstructure meets the volume fraction constraint.

[0044] S4, Multi-scale topology optimization process: Combining the surrogate model and finite element analysis, calculate the global stiffness matrix and structural compliance, and through iterative optimization by minimizing the loss function, obtain the optimal multi-scale structural topology that meets the volume constraint.

[0045] This step aims to achieve multi-scale structural design through iterative optimization, and the specific process is as follows:

[0046] Initial parameter setting: Discretize the design domain into grid cells (such as 100*100 grid), initialize the neural network weights, and generate the initial cutting height field.

[0047] Fast calculation of mechanical properties: In the process of calling the surrogate model, according to the cutting height parameters of each unit, predict its equivalent elastic matrix and volume fraction through the surrogate model. In the finite element analysis process, construct the element stiffness matrix based on the predicted elastic matrix and assemble it into the global stiffness matrix. By solving the equilibrium equation, obtain the displacement response of the structure.

[0048] Definition and optimization of the objective function: When designing the loss function, the structural compliance (the product of displacement and load) can be used as the main optimization objective, combined with the penalty term of the volume fraction constraint, to form a comprehensive loss function. For example, the total loss = structural compliance + λ * (actual volume fraction - target volume fraction) 2 , where λ is the weight coefficient, balancing the importance of compliance and volume constraint. The iterative optimization process is to update the neural network parameters through backpropagation, and loop through the process of "predicting mechanical properties → calculating loss → updating parameters" until the loss function converges.

[0049] Result output and application: Finally, output the cutting height field and map it to the multi-scale topological structure. This structure can be applied to additive manufacturing to ensure the continuous connectivity and mechanical properties of the microstructures.

[0050] In summary, the multi-scale structural stiffness optimization design method based on neural network reparameterization in the embodiments of the present invention encodes the design variables (cutting height parameters) in multi-scale topology optimization as network weights through neural network reparameterization technology, and uses automatic differentiation to achieve efficient calculation of gradients, avoiding the complex manual derivation in traditional sensitivity analysis. The introduction of the surrogate model significantly reduces the computational cost of homogenization analysis, enabling the optimization process to be completed with ordinary computing resources. The finally designed multi-scale structure has both high stiffness and lightweight characteristics and is suitable for the design of complex components in fields such as aerospace and automotive.

[0051] In some alternative embodiments of the present invention, the level set method adjusts the height of the cutting plane by fixing the level set function to generate microstructures with different volume fractions, and the range of its volume fraction is from 0.2 to 0.8.

[0052] It is understandable that the volume fraction and the cutting height are linearly related, and their sum is 1. For example, when the cutting height is 0.3, the volume fraction is 0.7, generating a high-density microstructure; when the cutting height is 0.7, the volume fraction is 0.3, generating a low-density microstructure.

[0053] In this embodiment, by limiting the range of the cutting height parameter (0.2 to 0.8), it is ensured that the volume fraction of the microstructure is within a reasonable range, balancing the mechanical properties and the lightweight requirements. When the cutting height is less than 0.2, the volume fraction of the microstructure is too low, which may cause the structure to break or be unable to effectively bear the load. When the cutting height is too high, the lightweight effect is significantly reduced.

[0054] In some alternative embodiments of the present invention, the proxy model construction step includes dataset generation: uniformly sampling the cutting height parameter through the homogenization theory to generate the elastic matrix and volume fraction data of the microstructure for training the proxy model.

[0055] Before constructing and training the proxy model, it is necessary to implement the generation of the dataset. In this embodiment, the generation process of the dataset includes the following steps:

[0056] Adopt the level set method to perform parametric modeling of the microstructure, control the height of the cutting plane to generate different volume fraction microstructure configurations;

[0057] Randomly sample the cutting height parameter, calculate the equivalent elastic tensor of the microstructure according to the homogenization theory, and obtain its mechanical property parameters;

[0058] Repeat multiple times to generate a dataset containing the cutting height parameter and its corresponding mechanical property parameters.

[0059] Designed in this way, high-precision data is generated through finite element analysis simulation, reducing the calculation cost, improving the prediction accuracy of the proxy model for the mechanical properties of the microstructure, and enabling the dataset to cover a wide range through random sampling, enhancing the generalization ability of the proxy model and improving the prediction accuracy.

[0060] In some alternative embodiments of the present invention, in the dataset generation step, the sampling interval of the cutting height parameter is uniformly and equally distributed, and the division ratio of the training set to the validation set is 9:1.

[0061] The sampling interval of the cutting height parameter adopts a uniformly and equally distributed manner to ensure that the dataset is evenly distributed in the entire design space. The dataset division ratio is set to 9:1, that is, 90% of the data is used for training to ensure that the model learns enough samples and improves the fitting ability. 10% of the data is used for validation to prevent the model from overfitting and improve the generalization ability.

[0062] In some alternative embodiments of the present invention, the surrogate model adopts a fully connected neural network, whose input is the cutting height parameter, and the output is the equivalent elastic matrix components and volume fraction of the microstructure, and the mean absolute error is used as the loss function for training. The fully connected neural network (FCNN), as a common neural network model, can handle the non-linear relationship between the input and output. In the present invention, the FCNN is used to establish the mapping relationship between the cutting height of the microstructure and the mechanical properties. The input layer of the FCNN is the cutting height parameter, the output layer is the mechanical property parameter, and the number of hidden layers and the number of neuron nodes are adaptively adjusted according to the scale of the data set. In the design of the neural network, the input layer receives the cutting height parameter, and the output layer predicts the mechanical property parameters of the microstructure. The number of hidden layers and the number of nodes of each layer of neurons will be adaptively adjusted according to the scale of the training data set to ensure that the model can accurately learn the relationship between the input and output.

[0063] In some alternative embodiments of the present invention, the output layer of the neural network optimizer adopts the Sigmoid activation function to limit the cutting height parameter within a preset range; the hidden layer adopts the LeakyReLU activation function to enhance the non-linear expression ability. The output layer adopts the Sigmoid activation function to limit the cutting height parameter h(x,y) within a reasonable range to ensure the physical feasibility of the optimization variables. The hidden layer adopts the LeakyReLU activation function to alleviate the vanishing gradient problem, enhance the non-linear expression ability of the neural network, and improve the optimization accuracy.

[0064] In some alternative embodiments of the present invention, the loss function in the multi-scale topology optimization process is defined as a weighted combination of the structural compliance and the volume fraction constraint, and the Adam optimizer is used to automatically calculate the gradient and update the neural network parameters.

[0065] Among them, the optimization goal is to minimize the structural compliance (i.e., maximize the stiffness) while satisfying the volume fraction constraint. Therefore, the loss function L is set as

[0066] L = C + λ(V - V*) 2

[0067] Among them, C is the structural compliance, which is used to measure the deformation degree of the structure under external force. The smaller the compliance, the greater the stiffness of the structure, and it is defined as: C = U T KU, where K is the global stiffness matrix, which is calculated by the element stiffness matrix K e through finite element analysis. U is the displacement vector, which represents the corresponding of the structure under the action of the external force F and satisfies the equilibrium equation: KU = F.

[0068] V is the current volume fraction, which is used to constrain the material usage of the structure during the optimization process. V *is the target volume fraction, which is set to the amount of material usage desired in the optimal design. λ is the volume fraction constraint factor, which is a weight parameter used to balance the structural compliance optimization and the volume fraction control.

[0069] In summary, by minimizing C, this loss function improves the structural stiffness and ensures that the optimization result has good mechanical properties. Through volume integration, the constraint term λ(V - V * ) 2 controls the material distribution, avoids too much or too little material inside during the optimization process, and ensures the manufacturability of the structure.

[0070] In the multi-scale topology optimization process, a neural network reparameterization optimization framework is adopted. The neural network parameters θ are used as design variables and optimized through automatic differentiation and backpropagation.

[0071] Among them, the forward propagation process includes the following steps carried out sequentially: taking the discrete element coordinates (x, y) in the design domain as the input of the neural network, and outputting the cutting height parameter h(x, y) through the neural network optimizer for microstructure generation. The equivalent elastic matrix C ij and the volume fraction V of the microstructure are calculated. The global stiffness matrix K and the displacement vector U are calculated. The loss function L is calculated.

[0072] The automatic differentiation process includes: calculating the gradient of the loss function L with respect to the neural network parameters θ through the automatic differentiation function of deep learning frameworks such as PyTorch and TensorFlow: Since L is calculated from the neural network output h(x, y), the automatic differentiation technique can be used to calculate the gradient without manually deriving the adjoint equation.

[0073] In the backpropagation and parameter update process, the Adam optimizer is used to update the neural network parameters:

[0074]

[0075] Among them, θ t is the neural network parameter after the t-th round of optimization; the learning rate η controls the update step size, and the gradient is calculated by automatic differentiation.

[0076] The Adam (Adaptive Moment Estimation) optimization algorithm is a commonly used gradient descent algorithm in the topology optimization process and is superior to the traditional SGD (Stochastic Gradient Descent) method. Its main features are as follows:

[0077] In some alternative embodiments of the present invention, in the multi-scale topology optimization process, the global stiffness matrix is obtained by combining the element stiffness matrices output by the surrogate model and is used to solve the displacement vector and the structural compliance.

[0078] Adaptive learning rate adjustment: Combining first-moment estimation (momentum) and second-moment estimation (exponentially weighted moving average of the squared gradient), it accelerates convergence in the early stage of optimization, reduces the learning rate in the later stage, and improves stability.

[0079] Applicable to sparse gradient optimization problems: The neural network optimizer of the present invention outputs the cutting height parameter h(x,y)h(x,y)h(x,y), and its gradient update is not uniform. Adam can improve the optimization stability.

[0080] The Adam optimization process is as follows:

[0081] (1) Calculate the first-moment estimation (gradient mean):

[0082] m t = β1m t-1 +(1 - β1)g t

[0083] where m t is the exponentially weighted average of the gradient, g t is the current gradient value, and β1 is the momentum parameter, with a default value of 0.9.

[0084] (2) Calculate the second-moment estimation (gradient squared mean): [[ID=X]]

[0085]

[0086] where v t is the exponentially weighted average of the squared gradient, and β2 has a default value of 0.999.

[0087] (3) Correct the bias

[0088]

[0089] (4) Update the parameters

[0090]

[0091] where is the corrected first-moment estimation (bias-corrected gradient mean), is the corrected second-moment estimation (bias-corrected gradient squared mean), m t is the uncorrected first-moment estimation, v t is the uncorrected second-moment estimation, β1 controls the exponential decay rate of the first momentum, with a default value of 0.9, β2 controls the exponential decay rate of the second momentum, with a default value of 0.999. t represents the current training step (iteration number), and ε is a small value, with a default of 10 -8 , used to avoid division-by-zero errors. Note: There seems to be an incomplete formula in the original text for calculating the second-moment estimation (the part marked as

[0085] in the original). I have translated it as "X" to maintain the integrity of the translation process. If you can provide the complete formula, I can further improve the translation.

[0092] In this embodiment, during the multi-scale topology optimization process, a weighted combination of structural compliance and volume fraction constraints is used as the loss function, and the Adam optimizer is used to automatically calculate the gradient and update the neural network parameters. This method has high optimization efficiency and strong stability. Finally, a multi-scale structure with high stiffness, reasonable material distribution, and suitable for additive manufacturing is obtained, which has obvious advantages compared with traditional optimization methods.

[0093] The multi-scale structural stiffness optimization design method based on neural network reparameterization provided by the embodiment of the present invention is applicable to additive manufacturing technology. The optimized multi-scale structure has non-uniform micro-space distribution and continuous connectivity. The embodiment of the present invention uses neural network optimization to make the microstructure distribution more flexible and improve the material utilization rate. Through the neural network optimizer, the situation of isolated or broken microstructures is avoided. By this method, the manufacturability of the optimized structure is improved, which is applicable to additive manufacturing processes such as 3D printing. The optimization results can be directly used for manufacturing without additional processing.

[0094] The following will detail each step and the related principles in the multi-scale structural stiffness optimization design method based on neural network reparameterization provided by the embodiment of the present invention.

[0095] I. Parametric modeling of microstructures

[0096] This method mainly uses the level set method for microstructure modeling. In the classical level set method, the structure boundary is implicitly represented by the zero level set of the high-dimensional level set function φ(x,t), and its geometric model can be expressed as:

[0097]

[0098] where x represents the coordinates of any point in the design domain D. represents the boundary of the structure domain Ω. t is a pseudo-time, which can represent the structure evolution process by driving the update of the level set function.

[0099] The level set method provides a flexible and convenient representation method for complex microstructures. In theory, the level set method allows the digital construction of any type of microstructure. When performing topology optimization using the conventional level set method, its optimization process fixes the zero horizontal plane, and then makes the boundary move continuously by evolving the level set function until the optimized structure is obtained. Another effective method is to fix the level set function and obtain different structural configurations by changing the cutting plane. Such as Figure 1As shown, a specific level set function is set, with the z-axis as the reference axis, and then the level set function is cut by cutting planes at different heights to obtain microstructures with different volume fractions. When the cutting height is h = -1, the volume fraction of the microstructure is 1. As the cutting height gradually increases, the volume fraction of the microstructure gradually decreases. When h = 1, the volume fraction of the microstructure is 0. The mathematical model of the microstructure configuration can be expressed as:

[0100]

[0101] where represents the pre-set level set function, C(x, h) represents the cutting plane, and h (-1 ≤ h ≤ 1) represents the height of the cutting plane. As Figure 3 shown, by setting different level set functions, different microstructure configurations can be obtained. It can be found from the figure that once the level set function is determined, only the cutting height h is the control parameter that determines the microstructure configuration.

[0102] II. Homogenization Analysis and Surrogate Model Construction

[0103] 2.1 Microstructure Homogenization Analysis

[0104] In the design of multi-scale structures, it is necessary to calculate the equivalent mechanical properties of microstructures at the macroscopic scale. Traditional homogenization theory is often used to calculate the equivalent properties of periodic composites and is widely applied in the field of multi-scale structure design. According to the homogenization theory, the homogenized elastic tensor of periodic microstructures can be calculated as:

[0105]

[0106] where |V m | represents the volume of the microstructure, C pqrs represents the local variable elastic tensor of the microstructure, represents the pre-set strain, represents the local strain. The local strain is defined as:

[0107]

[0108] where χ ij represents the generalized displacement, which satisfies the following equation:

[0109]

[0110] where v represents the virtual displacement field. For two-dimensional problems, the superscripts and subscripts (ij, pq, and kl) can be expressed as:

[0111]

[0112] For two-dimensional anisotropic microstructures, the equivalent elastic matrix C H has six independent components, namely C 11 , C 12 , C 13 , C 22 , C 23 and C 33 . The remaining three components are the same as the symmetric values in the matrix. However, when the microstructure has two perpendicular symmetry axes, the microstructure has isotropic mechanical properties. At this time, the equivalent elastic matrix C H has only three independent non-zero components, namely C 11 , C 12 and C 33 . At the same time, C 22 is equal to C 11 , C 21 is equal to C 12 , and the remaining components in the matrix are all 0.

[0113] 2.2 Construction of the surrogate model

[0114] In the optimization process of multi-scale topology optimization, it is very time-consuming to perform real-time homogenization analysis on the microstructure, and the computational cost is often unbearable. To overcome this problem, the present invention uses a neural network to learn the mapping relationship between the cutting height parameter and the mechanical properties of the microstructure. Subsequently, a high-precision surrogate model for quickly solving the mechanical properties of the microstructure can be obtained.

[0115] In this method, the cutting height h (-1 ≤ h ≤ 1) is sampled at uniform equal intervals Δk, and then 2 / Δk + 1 data samples can be obtained. The Young's modulus and Poisson's ratio of the material are set as E and μ respectively. During the homogenization analysis, the microstructure is discretized into a matrix of n = 100×100 elements, where 1 and 0 represent the presence of material (solid) and the absence of material (void) respectively. Among them, n1 and n0 represent the number of solids and voids respectively. Therefore, the volume fraction v e of the microstructure can be obtained from v e = n1 / n. During the data generation process, the elastic matrix of the microstructure is calculated through the numerical homogenization theory. Uniform sampling can ensure good uniformity of the cutting height parameter space. To ensure the effectiveness of the training process, this method randomly divides the data set into a training set (90%) and a validation set (10%).

[0116] Here, this method selects a fully connected neural network (FCNN) as the model to establish a high-precision surrogate model. The surrogate model constructs the mapping between the microstructure height parameter h and the corresponding performance parameters (C 11 , C 12 , C 13 , C 22,C 23 ,C 33 ,v e ) between the mappings, so take X = h as the input of the surrogate model, and take Y = [C 11 ,C 12 ,C 13 ,C 22 ,C 23 ,C 33 ,v e T as the network output. The mean absolute error (MAE) is used as the loss function for model training, and its definition is as follows:

[0117]

[0118] In the formula, is the label value obtained by the homogenization method. yi and represent the components of vector Y and respectively. Through model training, a high-precision surrogate model is obtained.

[0119] III. Neural Network Architecture

[0120] Neural networks are often used to learn the internal laws between existing data in many cases, facilitating the construction of surrogate models to quickly solve new problems or designs. In addition, neural networks can also be used as optimizers to replace optimization algorithms, with the main difference lying in the construction of the model loss function. The optimization mechanisms in both usage scenarios use the gradient descent method to adjust the internal parameters of the neural network to minimize the loss function.

[0121] The present invention develops a reparameterized optimizer for neural networks. Figure 4 The overall network architecture of the present invention is given, which is easy for everyone to understand intuitively. Figure 4 (a) shows the neural network structure used for reparameterization in this study, which is also the part that this method focuses on and studies. And Figure 4 (b) shows the network structure of the surrogate model, mainly used for the rapid solution of the mechanical properties of microstructures. It can be observed from Figure 4 (a) that each connection in the neural network is associated with a weight, and each node is associated with an activation function and a bias. The input of the reparameterized network is the coordinates (x, y) of each point in the structural design domain, and the output is the initial cutting height parameter of the microstructure As Figure 4 (a) shows, as an example, calculate the value of to be Then, the output of the node is calculated as where g f represents the activation function. Therefore, the output of the i-th neuron in the l-th layer​ The calculation is as follows:

[0122]

[0123] where represents the weight associated with the \(j\)-th neuron in the previous layer \(l - 1\) connected to the \(i\)-th neuron in layer \(l\), represents the bias associated with the \(i\)-th neuron. In this study, LeakyRelu is used as the activation function for the hidden layer. Additionally, to ensure that the initial cutting height is within the range of \([0, 1]\), the present invention selects Sigmoid as the activation function for the output layer. The expression of the Sigmoid activation function is as follows:

[0124]

[0125] It should be noted that the value range of the cutting height of the parametric microstructure is \([-1, 1]\). Therefore, it is necessary to convert the initial cutting height to the cutting height. The conversion formula is as follows:

[0126]

[0127] The obtained cutting height \(h\) is used as Figure 4 the input of the surrogate model in (b), and then the mechanical properties of the microstructure are solved.

[0128] IV. Description of the multi-scale topology optimization problem

[0129] In the present invention, the classical multi-scale topology optimization problem is mainly considered, with particular attention paid to the optimization of the structural stiffness under the linear elastic model. Taking this problem as an example, the basic solution process of multi-scale topology optimization is illustrated. In the process of parametric modeling of the microstructure, the cutting height parameter is used as the variable to control the geometry of the microstructure. Therefore, the multi-scale topology optimization problem can be expressed as:

[0130]

[0131] where \(C\) represents the structural compliance, \(H\) represents the global cutting height vector, \(F\) represents the applied load vector, \(K\) represents the overall stiffness vector, \(U\) represents the displacement vector, \(N\) represents the number of discrete elements of the macrostructure, \(v\) e represents the element volume fraction, \(V\) * represents the desired volume fraction. Here, the volume fraction \(V\) * is limited to the range of 0.2 to 0.8. The lower limit design of \(V\) * is to prevent the disconnection phenomenon of the microstructure caused by too small a volume fraction. At the same time, the upper limit of \(V\) * is defined to prevent the entire design area from being completely filled with materials due to too large a volume fraction, resulting in no microstructure configuration. \(v\) minis a small positive value, serving as the lower bound of the volume fraction to avoid the singularity of calculating the global stiffness matrix. h min is the minimum value of the cutting height parameter, ensuring that the volume fraction of the microstructure does not exceed 0.8. h min The value of depends on the type of microstructure.

[0132] Once the design problem is determined, the corresponding boundary conditions and neural network inputs are established. The design variable h now mainly depends on the parameters θ of the neural network. In this case, optimizing the loss function of the reparameterized network enables obtaining the optimal parameters θ, and thus the optimal h. However, the goal of neural network optimization is to minimize the loss function, which is an unconstrained optimization problem. Therefore, it is necessary to transform the constrained minimization problem into an unconstrained optimization problem. Then, the loss function is defined as:

[0133]

[0134] where θ represents the parameters of the NN, including the weights w and the biases b. α represents the penalty parameter, and C0 represents the initial compliance of the optimization, i.e., the compliance calculated for the first time. The final solution is obtained by minimizing the loss function. During the optimization process, the neural network automatically calculates the gradients and performs backpropagation to update the parameters θ of the neural network. Then, the above steps are iteratively repeated until the termination condition is satisfied, thereby obtaining the optimal multi-scale topology.

[0135] V. Multi-scale Structural Optimization Process Based on Neural Network Reparameterization

[0136] The process of this method is mainly divided into four parts, namely: dataset generation, surrogate model construction, neural network reparameterization, and optimization, as follows Figure 5 shown.

[0137] Dataset generation: Determine in advance the size of the design domain and the material condition parameters, and use the numerical homogenization method to calculate the elastic matrix of the microstructure. By uniformly sampling and changing the cutting height, use the level set method to generate a specific number of datasets as the training data for surrogate modeling. To ensure a high-precision surrogate model, the data samples are distributed as evenly as possible in the design space.

[0138] Surrogate model construction: The cutting height h is used as the input of the neural network surrogate model, and the elastic matrix components and the microstructure volume fraction calculated through the homogenization theory are used as the training labels of the model. The neural network learns the inherent laws of the existing data and then serves as a surrogate model to quickly predict new data. The mapping modeling process is modeled as a regression task, and by adjusting the hyperparameters, a high-precision surrogate model is obtained through model training.

[0139] Neural network reparameterization: First, this method establishes a neural network for reparameterization, denoted as fθ Here, the size of the neural network needs to be set, including the depth (D) and width (W). The depth of the neural network represents the number of layers of the network model, and the width of the neural network represents the number of neuron nodes in each layer. The input of the network is the coordinates (x, y) of each discrete unit in the design domain, and the output is obtained through the forward propagation of the network f θ The number of inputs and outputs is determined by the grid discretization of the design domain. For example, if the design domain is discretized using a 30×60 grid, the dimension of the input vector is 2×1800, and the dimension of the output vector is 1×1800. In this process, the independent variable is the parameter of the neural network f θ and the initial cutting height output by the neural network is the dependent variable. It converts the direct optimization of the cutting height of the level set method into the update of the parameters of f θ (such as weights w and biases b). This conversion and update of variables is the main feature of neural network reparameterization.

[0140] Optimization: As Figure 5 shown, for the cutting height output by f θ this method calls the surrogate model to quickly solve the elastic matrix C H and the microstructure volume fraction v e . Then, according to C H the element stiffness matrix k is calculated through finite element analysis e , and the element stiffness matrices k e are combined into the global stiffness matrix K. The displacement vector U is calculated by combining the macroscopic boundary constraints and the applied force F. According to the global stiffness matrix K and the displacement vector U, the compliance of the structure can be calculated using C = U T KU. At the same time, the microstructure volume fractions v e are added up to obtain the total volume fraction V. Then the objective function and the volume fraction are combined to calculate the loss function. Secondly, the Adam optimizer is used to automatically calculate the gradient (sensitivity) of the loss function and backpropagate to update the neural network parameters. During the entire optimization process, the above steps are iteratively repeated until a stable multi-scale topological configuration is obtained and the optimization iteration terminates.

[0141] ​It should be noted that in this text, relational terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the term "comprising", "including" or any other variant thereof is intended to cover non-exclusive inclusion, so that a process, method, article or device comprising a series of elements not only includes those elements, but also includes other elements not expressly listed, or further includes elements inherent to such process, method, article or device. In addition, in this text, "front", "rear", "left", "right", "upper" and "lower" are all referenced with respect to the placement state shown in the drawings.

[0142] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A multi-scale structural stiffness optimization design method based on neural network reparameterization, characterized in that Including: Microstructure parametric modeling: The level set method is used to perform parametric modeling on the microstructure, and the volume fraction and geometric configuration of the microstructure are controlled by adjusting the cutting height parameter; Surrogate model construction: A neural network is used to establish the mapping relationship between the cutting height parameter and the equivalent mechanical properties of the microstructure, forming a high-precision surrogate model for quickly predicting the elastic matrix and volume fraction of the microstructure; Neural network reparameterization design: A neural network optimizer is constructed, taking the coordinates of each discrete unit in the design domain as the input and outputting the corresponding cutting height parameter; The neural network parameters are updated through automatic differentiation and backpropagation to achieve continuous optimization of the design variables; Multi-scale topology optimization process: Combining the surrogate model and finite element analysis, calculate the global stiffness matrix and structural compliance, and iteratively optimize through minimizing the loss function to obtain the optimal multi-scale structure topology that meets the volume constraint.

2. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, characterized in that The level set method generates microstructures with different volume fractions by fixing the level set function and adjusting the height of the cutting plane, and the volume fraction range is from 0.2 to 0.

8.

3. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, wherein The steps of constructing the surrogate model include data set generation: Uniform sampling of the cutting height parameter is performed through the homogenization theory to generate the elastic matrix and volume fraction data of the microstructure for training the surrogate model.

4. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 3, characterized in that In the data set generation step, the sampling interval of the cutting height parameter is uniformly equally spaced, and the division ratio of the training set to the validation set is 9:

1.

5. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, characterized in that The surrogate model uses a fully connected neural network, with the cutting height parameter as the input, the equivalent elastic matrix and volume fraction of the microstructure as the output, and the mean absolute error as the loss function for training.

6. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, characterized in that The output layer of the neural network optimizer uses the Sigmoid activation function to limit the cutting height parameter within a preset range; The hidden layer uses the LeakyReLU activation function to enhance the non-linear expression ability.

7. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, characterized in that In the multi-scale topology optimization process, the loss function is defined as the weighted combination of the structural compliance and the volume fraction constraint, and the Adam optimizer is used to automatically calculate the gradient and update the neural network parameters.

8. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, characterized in that In the multi-scale topology optimization process, the global stiffness matrix is obtained by combining the element stiffness matrices output by the surrogate model and is used to solve the displacement vector and structural compliance.

9. The multi-scale structural stiffness optimization design method based on neural network reparameterization according to claim 1, wherein The method is applicable to additive manufacturing technology, and the optimized multi-scale structure has non-uniform micro-space distribution and continuous connectivity.

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