Efficient Complex Domain Analysis and Topology Optimization Methods Based on Problem-Free Transfer Learning

Abstract

This invention belongs to the field of machine learning and topology optimization technology, specifically involving a method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning. The method includes generating samples for training a machine learning model; constructing and training a source task neural network to predict the numerical shape functions of regular substructures; constructing and training a target task neural network to predict the numerical shape functions of isoparametric substructures through transfer learning, thus obtaining a problem-independent transfer learning model; predicting the numerical shape functions of each substructure based on its material distribution and node coordinates; assembling the substructures to form a total stiffness matrix; and updating the design variables using an optimizer. This invention overcomes the limitation of existing topology optimization methods that are only applicable to regular design domains, reuses pre-trained models to reduce training costs, embeds mechanical information to ensure analytical accuracy, significantly improves the optimization efficiency of complex design domains, has strong model versatility, adapts to various engineering scenarios, and has promising application prospects.

Description

Technical Field

[0001] This invention belongs to the field of machine learning and topology optimization technology, specifically involving a method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning. Background Technology

[0002] Structural topology optimization aims to find the optimal material layout through rational means, so as to achieve the best performance of the structure under given constraints. However, topology optimization involves the iterative solution of partial differential equations with varying coefficients, and for high-resolution topology optimization problems in complex design domains, the high computational cost is often prohibitive. The rise of deep learning technology has brought new opportunities for the development of topology optimization. The universal approximation property and high computational efficiency of deep neural networks provide an effective means to solve the efficiency problem of topology optimization. Many scholars have attempted to construct global proxy models for topology optimization based on end-to-end mapping. While this can achieve near real-time topology optimization design, it also has problems such as difficulty in constructing datasets and difficulty in guaranteeing the generalization performance of the model. To address the aforementioned issues, some scholars have proposed a problem-agnostic machine learning model. This method, based on the substructure method, establishes an implicit mapping between the material distribution of substructures and their numerical shape functions through a machine learning model, thereby improving the computational efficiency of topology optimization. However, existing research primarily focuses on relatively regular design domains, analyzing and optimizing based on regular quadrilateral and hexahedral elements. For the numerous three-dimensional complex design domain problems encountered in practical engineering, introducing isoparametric elements that more accurately describe geometric boundaries is undoubtedly more appropriate. Therefore, there is an urgent need for a problem-independent transfer learning structure topology optimization method applicable to complex design domains, in order to reduce the training cost of machine learning models, improve the generalization ability of models, and achieve efficient solutions to large-scale structure topology optimization problems in complex design domains. Summary of the Invention

[0003] The purpose of this invention is to provide an efficient analysis and topology optimization method for complex domains based on problem-independent transfer learning. This invention overcomes the limitation of existing topology optimization methods that are only applicable to regular design domains, reuses pre-trained models to reduce training costs, embeds mechanical information to ensure analysis accuracy, significantly improves the optimization efficiency of complex design domains, has strong model versatility, is adaptable to various engineering scenarios, and has good application prospects.

[0004] The specific technical solution adopted by this invention is as follows: An efficient analysis and topology optimization method for complex domains based on problem-independent transfer learning includes the following steps: S1: Generate samples for training machine learning models; S2: Construct and train the source task neural network for predicting the numerical shape functions of the rule substructure; S3: Based on the source task neural network, a target task neural network for predicting isoparametric substructure numerical shape functions is constructed and trained through transfer learning to obtain a problem-independent transfer learning model. S4: In the topology optimization process, the trained problem-independent transfer learning model is used to predict the numerical shape function of each substructure based on the material distribution and node coordinates. S5: Calculate the outlet stiffness matrix of each substructure based on the predicted numerical shape function, and assemble them to form the total stiffness matrix; S6: Update the design variables using the optimizer, iterate until convergence, and obtain the optimal topology.

[0005] The technical effects achieved by this invention are as follows: This invention is the first to apply problem-independent transfer learning to topology optimization in complex 3D design domains, breaking through the limitation of existing methods that are only applicable to regular design domains. It expands the application scope of problem-independent machine learning methods in complex engineering structures. By reusing pre-trained models of regular substructures, it significantly reduces the cost of dataset construction and model training, greatly improving model training efficiency. Furthermore, the neural network structure embedded with mechanical information ensures the completeness of the displacement field and the rank preservation of the substructure exit stiffness matrix, theoretically improving the accuracy of finite element analysis and enhancing the reliability of topology optimization results. By accelerating the finite element analysis process with the help of machine learning models, it significantly reduces the overall computation time of topology optimization and significantly improves the solution efficiency of large-scale structural topology optimization in complex design domains. The established model is independent of specific optimization problems, design domains, boundary conditions, and load conditions, and can be adapted to different engineering scenarios without readjustment or training. It has shown good accuracy and stability in a variety of typical topology optimization problems and has excellent engineering application prospects. Attached Figure Description

[0006] Figure 1 This is a schematic diagram of a sample generated by connecting eight randomly generated points according to the present invention; Figure 2 This is a schematic diagram of the generation process and sample using the six-face intersection method in this invention; Figure 3 This is a schematic diagram of the problem-independent transfer learning network structure in this invention; Figure 4 This is a schematic diagram of the neural network, mechanical information, and the embedded components in this invention; Figure 5 This is a schematic diagram comparing transfer learning and non-transfer learning in this invention; Figure 6 This is a schematic diagram illustrating the performance of the machine learning model in this invention on the test set; Figure 7This is a schematic diagram of the bracket problem (flexibility minimization problem) in this invention; Figure 8 This is a schematic diagram of the optimization results and objective function for the minimum compliance problem in this invention; Figure 9 This is a schematic diagram illustrating the iterative history of the minimum compliance problem in this invention; Figure 10 This is a schematic diagram illustrating the optimization time for the minimum compliance problem in this invention; Figure 11 This is a schematic diagram of the compliant mechanism problem in this invention; Figure 12 This is the iterative history of the compliant mechanism problem in this invention; Figure 13 The problem of compliant mechanisms in this invention lies in three body ratio constraints ( Optimization results under ( ) Figure 14 This is the iterative history of the compliant mechanism problem under different volume fraction constraints in this invention; Figure 15 This is a flowchart of the calculation process in this invention. Detailed Implementation

[0007] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0008] In the field of engineering, there are numerous complex structural analysis and optimization design problems. To accurately describe the geometric boundaries of a structure, high-resolution finite element substructures are required. The resulting enormous computational overhead becomes a bottleneck for the efficiency of complex engineering structural analysis and optimization design. In the three-dimensional context, real structures often contain a large number of isoparametric elements with varying shapes. Taking a three-dimensional hexahedral isoparametric element as an example, to build a dataset of hexahedral isoparametric element substructures, it is necessary to generate hexahedrons of various shapes. This not only requires the generated hexahedrons to cover as many real hexahedral structures as possible, but also means that training neural networks is extremely costly. In other words, three key issues need to be addressed. First, how to efficiently generate substructures with varying shapes, while ensuring that the elements composing the substructures satisfy displacement compatibility. Second, how to build a lightweight, low-training-cost machine learning model that not only has the advantages of high inference efficiency and the ability to quickly predict a large number of substructure numerical shape functions, but also has low training costs, allowing for the training of a high-accuracy machine learning model with only a few samples. Third, how to ensure the high accuracy of the predicted numerical shape functions, thereby guaranteeing the accuracy of finite element analysis. To address the above problems, this invention will introduce a problem-independent transfer learning model based on the embedding of mechanical information; like Figures 1-15 As shown, the efficient analysis and topology optimization method for complex domains based on problem-independent transfer learning includes the following steps: S1: Generate samples for training machine learning models; To train a machine learning model, a large number of isoparametric substructures with different shapes and material distributions need to be generated. Taking a hexahedral eight-node isoparametric element as an example, if samples are generated by connecting eight randomly generated points, the following problems exist: First, it cannot be guaranteed that the nodes are coplanar, such as... Figure 1 As shown in (a), displacement compatibility is difficult to guarantee; secondly, the sample may exhibit excessive distortion, such as... Figure 1 As shown in (b), (a) the nodes are not coplanar; (b) the structure is excessively distorted; and it is difficult to find an effective screening mechanism to control sample quality. To address these issues, this invention employs a six-face intersection method to generate samples. A point with a distance of 1 from the origin is generated along the principal axis of the Cartesian coordinate system. Then, a vector is randomly generated and used as the normal vector. This point uniquely determines a plane, such as... Figure 2 As shown in (a), repeating this process along all the positive and negative semi-axes of the coordinate system will yield six planes. If the control normal vectors vary randomly within a certain range, these planes will intersect each other to form a hexahedron, as shown in (a). Figure 2 As shown in (b), the hexahedron is subdivided at a given resolution and filled with random material to obtain a substructure. Finally, one node of this substructure is translated to the origin, and the coordinates of all nodes are normalized. This yields a sample, as shown in [example image]. Figure 2 As shown in (c). A large number of high-quality samples can be obtained simply by appropriately changing the normal vector, such as... Figure 2 As shown in (df), (ac) represents the sample generation process; (df) represents the sample generated by the hexagonal intersection method. These hexahedral isoparametric substructures generated by the hexahedral intersection method have coplanar boundary nodes, thus naturally satisfying displacement compatibility. This method has the advantages of low computational cost and no need for sample selection; a large number of substructures with different shapes can be generated in a short time simply by changing the normal vector. Furthermore, since the substructure is independent of the design domain and boundary conditions of the optimization problem, the numerical shape function of the substructure is uniquely determined given the coordinates of its eight vertices and the material distribution. The sample generation process and the machine learning process are independent of the optimization problem; therefore, these processes are all performed offline. S2: Construct and train the source task neural network for predicting the numerical shape functions of the rule substructure; Although the six-face intersection method generates a large number of isoparametric substructures, the substructures in real-world problems often have very complex shapes and varying material distributions. Only when the number of samples is sufficiently large, even approaching infinity, will isoparametric substructures with specific shapes or material distributions appear in the training set. A massive training set undoubtedly increases the training cost of machine learning models significantly. The difference between isoparametric substructures and regular substructures lies only in the node coordinates; the calculation process of the numerical shape function is essentially the same. If the knowledge learned in the machine learning model predicting the numerical shape function of regular substructures can be applied to the prediction of the numerical shape function of isoparametric substructures, instead of training a new model from scratch, it might reduce the requirement for a large number of samples and lower the training cost. To achieve this, the concept of transfer learning is introduced here.

[0009] Transfer learning aims to apply knowledge or patterns learned from past tasks (source tasks) to different but related new tasks (target tasks), thereby reducing training costs and accelerating the training process. In this invention, predicting the numerical shape function of the rule substructure is considered the source task. Based on the source task model, an additional neural network is added and the output of the source task network is fine-tuned to achieve the target task of predicting the numerical shape function of the isoparametric substructure. The details of the machine learning model will be introduced below; First, we introduce the source task, which is a machine learning model for predicting the numerical shape function of a regular substructure. The input to this model consists of two parts: the material distribution of the substructure and the coordinates of a detail point. Given the material distribution of a regular substructure and the coordinates of a detail point, the output is the numerical shape function corresponding to that detail point. The neural network structure of the source task includes a first branch network and a one-layer backbone network, each composed of a multilayer perceptron, as shown below. Figure 3 The blue boxes show (a) the original neural network; (b) the mechanical information; and (c) the neural network with embedded mechanical information.

[0010] The formula for calculating the first branch network is: (1) The input layer of the first branch network is the first... Material distribution of regular substructures, i.e. This represents the density vector of the fine elements of the constituent substructure. This represents the activation function. These represent the first branch network, respectively. The weights, biases, and output values ​​of the hidden layers; the number of hidden layers in the first branch network is... .

[0011] The formula for calculating the backbone network is: (2) (3) Among them, the input layer of the backbone network is the first... The first rule substructure The node coordinates of the detail point, i.e. ; coordinates of detail points All terms expanded using first, second, and third-degree complete polynomials are used as neurons in the actual input layer, i.e. ; These represent the backbone network's... The weights, biases, and output values ​​of the hidden layers, and the number of hidden layers in the backbone network are as follows: ; The output layer is obtained by performing a Hadamard product on the last hidden layers of the branch network and the backbone network. The weights and biases of the output layer are respectively used This indicates that, in the source task, the output layer is the first rule substructure. Numerical functions for detail points.

[0012] S3: Based on the source task neural network, a target task neural network for predicting isoparametric substructure numerical shape functions is constructed and trained through transfer learning to obtain a problem-independent transfer learning model. For the related but distinct novel problem of predicting numerical shape functions of isoparametric substructures, simply replacing the input of the backbone network with the coordinates of detail points of the isoparametric substructures and training the model from scratch not only requires generating a large number of new samples, resulting in extremely high training costs, but may also waste already learned knowledge. Transfer learning aims to leverage previously learned knowledge and experience to more effectively learn related but novel concepts, thereby reducing training time and data requirements. Therefore, to achieve transfer learning from predicting numerical shape functions of regular substructures to predicting numerical shape functions of isoparametric substructures, this invention adds an additional branch network, namely the second branch network, to the source task network, as follows: Figure 3 As shown in the red box, this second branch network is still a multilayer perceptron; S3 specifically includes: The input to the second branch network is the coordinates of the exit node of the isoparametric substructure. The calculation formula is as follows: (4) These represent the additional branch network's first branch, the second branch, and the third branch, the fourth branch, the fifth branch, and the sixth branch, ... seventh branch, and the sixth branch, respectively. The weights, biases, and output values ​​of the hidden layers, and the number of hidden layers are used. express; This will be the last hidden layer of the second branch network. Output layer of the original backbone network Perform the Hadamard product as the final output layer. And it is used to predict the numerical shape function of isoparametric substructures, that is: (5) The network of the source task ( Figure 3 The blue portion (as shown in blue) has been pre-trained, and its weights and biases remain unchanged during the training of the proposed model. Only the target task network (…) Figure 3 The weights and biases (in red) are updated during training. Thus, while fully utilizing the knowledge from the source task, an additional multilayer perceptron is added, and the output of the source task is corrected using the Hadamard product, thereby completing the target task, namely predicting the numerical shape function of the isoparametric substructure.

[0013] During training, only the weights and biases of the second branch network are updated, and the loss function is the numerical shape function prediction. Compared with the true value The mean squared error (MSE) between the two is calculated using the following formula: (6) Since the numerical shape function of the substructure depends only on the material distribution and geometry of the substructure, and is independent of the design domain, boundary conditions, and loads, this machine learning model is problem-independent and can be trained offline. When solving the topology optimization problem, the pre-trained model is directly loaded without any additional training.

[0014] S4: In the topology optimization process, a trained problem-independent transfer learning model is used to predict the numerical shape function of each substructure based on the material distribution and node coordinates. When predicting numerical shape functions in S4, the network output is constrained based on the invariance derived from mechanical principles to embed mechanical information and ensure the completeness of the displacement field.

[0015] Although substructures with different material distributions and shapes have different numerical shape functions, these functions possess a certain invariance determined by mechanical principles. However, neural networks exhibit probabilistic convergence. If no constraints are imposed on the neural network, the predicted numerical shape functions will not satisfy the invariance. Consequently, the substructure exit stiffness matrix calculated subsequently will be rank-insecure, and the completeness of the displacement field cannot be guaranteed. In this invention, the "invariance" derived from mechanical information (mechanical principles) is treated as a constraint and added to the network's output layer, constructing a neural network with embedded mechanical information. Compared to the original network, this network's output layer not only has fewer neurons, but the predicted numerical shape functions also satisfy the invariance. The exit stiffness matrix calculated from this shape function is rank-insecure, thus ensuring the completeness of the displacement field. The network and its theoretical foundation will be described in detail below. Under the assumption of small deformation, an elastic body does not generate strain energy when it undergoes rigid body displacement. Therefore: (7) (8) in, This represents the total stiffness matrix obtained by assembling the element stiffness matrices of all the fine elements in the substructure. This represents the outlet stiffness matrix of the substructure. Rigid body displacement mode representing detail points The rigid body displacement modes representing the substructure exit nodes are as follows: (9) (10) (11) in, Each element represents 1, and the dimension is 1. The matrix. represent The identity matrix. This indicates the translational displacement mode. This represents the rotational displacement mode. To simplify calculations, All are set to 1. Indicates the first The coordinate vector of the exit node of the substructure. Indicates the first The coordinate vectors of all detail points of the substructure. The number of exit nodes representing a single substructure, in this invention . This represents the number of detail points contained in a single substructure. Therefore, the numerical shape function of the substructure possesses the following invariance: (12) (13) It can be arranged according to the positional relationship of detail points. Block by line These are, in order, the numerical shape functions for boundary detail points and interior detail points. In this invention, due to the use of linear displacement interpolation, therefore... Similarly, the rigid body displacement mode of detail points can be... Similarly, it is divided into blocks. It is easy to know that... , This represents the number of detail points within the substructure. If used... To represent the resolution of the substructure, then ; If we directly use the numerical shape functions predicted by the neural network in subsequent calculations, without adding any constraints, such as Figure 4 As shown in (a), due to the probabilistic convergence of neural networks, the predicted numerical shape function will not strictly satisfy the invariance shown in Eq. (1.10b). Rigid body displacement (such as...) Figure 4 (b) As shown, the constant strain displacement state cannot be obtained, and the completeness of the displacement field function cannot be guaranteed. This will directly affect the accuracy of the finite element analysis. Figure 4 (a) Transfer learning; (b) No transfer learning; The process of embedding mechanical information is as follows: Numerical shape functions of detail points inside the substructure Block by column and The rigid body displacement mode of the substructure exit node Block by line and ,as follows: (14) (15) for Obtained by dividing into blocks by line. Substituting Eq.(1.11a) and Eq.(1.11b) into Eq.(1.10b), only... As part of neural network prediction The following formula is used to calculate: (16) Among them, only As part of neural network prediction No longer output by a neural network, but based on Therefore, it is calculated that... Treat it as a predicted value, Treat it as a calculated value, derived from the predicted value. Compared with calculated values Numerical shape functions Since invariance is satisfied, the completeness of the displacement field function is guaranteed, the substructure outlet stiffness matrix is ​​rank-preserving, and the accuracy of finite element analysis is improved.

[0016] S5: Calculate the outlet stiffness matrix of each substructure based on the predicted numerical shape function, and assemble them to form the total stiffness matrix; The proposed machine learning model was built using Tensorflow 2.6, and the model architecture is shown in Table 1. The number of hidden layers in each network is... The backbone network has an additional output layer, totaling... Branch network 2 has an additional output layer, for a total of... Layers. A total of 500,000 isoparametric substructures were generated to train the model. The batch size was 200, the iteration per epoch was 2500, and the epoch was 10. Automatic differentiation (tensorflow.GradientTape) was used to calculate the gradient of neurons. Adam was used as the optimizer to update the weights and biases of the network.

[0017] The table above is Table 1, which shows the PITL model architecture; The differences between the transfer learning model proposed in this invention and the non-transfer learning model are as follows: Figure 5 As shown. The black "NoTransfer" indicates direct training: the weights and biases of all networks (backbone network, branch network 1, branch network 2) are assigned random initial values ​​and continuously updated during training; the red "Transfer" indicates the proposed PITL model: the source task network has been pre-trained, and its weights and biases are directly loaded into this transfer learning model and remain unchanged during training. Only the weights and biases of the target task network change during training. It is worth noting that the "NoTransfer" model in the initial stage ( The model exhibits a lower loss value. This counterintuitive phenomenon stems from its random uniform initialization mechanism: the predictions generated by this model generally have small amplitudes and are statistically centered around the true values. In contrast, the "transfer" model is initialized using a feature extractor for regular substructures. When this pre-stored pattern acts together with the uninitialized branch network 2 on the isoparametric input, its pointing direction is biased, leading to temporary region mismatch and higher initial error.

[0018] However, as training progresses, the loss function of "Transfer" decreases rapidly. It converges to a specified threshold. "NoTransfer" still requires significant additional training. The left and right loss functions only reach similar levels. This demonstrates that the proposed transfer learning model requires fewer samples and less training time to achieve the same loss function value.

[0019] To evaluate the performance of the transfer learning and non-transfer learning machine learning models, a total of 10,000 samples were generated as a test set. The transfer learning model employs... The training results, the non-transfer model adopts The training results. The test results are as follows: Figure 6 As shown in the figure, the predicted values ​​of the numerical shape function are generally consistent with the actual values, but some errors still exist. However, the introduction of the mechanical information neural network ensures the completeness of the displacement field function, making the substructure exit stiffness matrix rank-preserving. This theoretically guarantees the accuracy of the finite element analysis, demonstrating that incorporating mechanical information is essential when constructing machine learning models.

[0020] S6: Update the design variables using the optimizer, iterate until convergence, and obtain the optimal topology.

[0021] Given the material distribution and node coordinates of isoparametric substructures, a problem-independent transfer learning model based on mechanical information embedding can quickly predict the numerical shape function of each substructure. Therefore, the proposed model can fully leverage the advantages of the substructure method in improving the computational efficiency of finite element analysis, enabling rapid analysis and topology optimization of high-resolution complex design domain structures. The output of this transfer learning model is a problem-independent numerical shape function, independent of the optimization method. This invention employs the SIMP method, a popular optimization technique in topology optimization. The efficiency and accuracy of the proposed model are demonstrated through a compliance minimization problem, and the generalization and effectiveness of the proposed method in solving non-self-adjoint problems are demonstrated through a compliant mechanism problem. The two problems differ only in their objective functions; the constraint functions are the same. The corresponding optimization formulations and numerical examples are shown below. The optimizer in step S6 is either the optimality criterion method or the moving asymptote method; the topology optimization problem is either the compliance minimization problem or the compliant mechanism problem. The optimization formula for the compliance minimization problem is: (17) Among them, the design variable is the element density, which is determined by... This indicates that the total number of fine units is determined by... express, This represents the total number of substructures. If the substructure resolution is... Then there is , Let these represent the objective function and the constraint function, respectively. This represents the exit stiffness matrix of each substructure in the set. The obtained total stiffness matrix, These are the substructure displacement vector and load vector, respectively. For material volume, For the first The volume of fine units, This indicates the upper limit of the volume fraction.

[0022] The SIMP interpolation model is used, with a penalty power. At the same time, introduce To prevent the overall stiffness matrix from becoming singular. Then the... Young's modulus of fine elements for: (18) (19) in, This represents a density vector based on density filtering. , Representing the The distance between fine units Indicates the density filtration radius. The weight coefficient matrix; To visually demonstrate the efficiency advantages of the proposed method, a bracket example is used to explore the impact of different substructure resolutions on efficiency. Considering... Figure 7 (a) illustrates a bracket problem where the blue area is fixed and the red column is subjected to a downward vertical load. The orange area represents the undesignable region, such as... Figure 7 As shown in (b), a volume fraction upper limit of 0.12 was set, and topology optimization with the objective of minimizing flexibility was performed on the structure. OC (The Optimality Criteria) was used as the optimizer. Half of the structure was optimized and symmetric boundary conditions were applied. A total of 250w fine meshes were created, and various methods were employed sequentially. Two substructures at different resolutions were calculated, with density filtering radii set to 3 and 7 times the average size of the finer mesh, respectively. The convergence condition was set to occur when the relative change in the objective function value over the last five consecutive iterations is less than [a certain value]. The optimization process terminates at that time, where, Figure 7 (a) Load diagram; (b) Diagram of undesignable region; The optimization result and objective function of this problem are as follows: Figure 8 As shown, the density cloud map indicates a threshold of 0.12. Figure 8 (a) is The optimization results of resolution machine learning, the objective function (compliance). The structure was reanalyzed using the substructure method with linear interpolation of boundary displacements to obtain the baseline objective function value. The relative error is 1.2%; The optimization results and objective function values ​​of resolution machine learning are as follows: Figure 8 As shown in (b), the objective function The relative error of the objective function was 5.9%. Furthermore, the relative errors between the displacement field obtained from the machine learning solution and the benchmark solution were 1.5% and 6.5%, respectively. In both substructure resolutions, the structure formed a reasonable force transmission path from the constraint end to the load end, verifying the accuracy of the proposed machine learning model. Next, OptiStruct was used to optimize the model with the same number of fine meshes, and the optimization results are as follows: Figure 8 As shown in (c), the objective function ,in, Figure 8 (a) based on (a) Machine learning optimization results based on resolution; (b) Based on (c) Machine learning optimization results for resolution; OptiStruct optimization results.

[0023] With the same total number of fine meshes, the proposed machine learning model can obtain an objective function similar to OptiStruct. Furthermore, the optimal topology and force transmission paths of both are similar, validating the effectiveness of the proposed method. The iteration history is as follows: Figure 9 As shown; Figure 10 The optimization time for this problem is shown. It's worth noting that OptiStruct is configured for 32-core parallel computation, while computation using the proposed machine learning model is all serial. The time consumption of a single optimization iteration step is shown below. Figure 10 As shown in (a), the average single-step optimization time (serial) for the two resolutions of machine learning is 63.54s and 116.90s, respectively, while the average time of OptiStruct (parallel 32 cores) is 8223.03s. That is, in the serial case, the proposed method still achieves a 129-fold and 70-fold efficiency improvement in single-step optimization time compared to OptiStruct parallel 32 cores. Furthermore, in terms of total optimization time, OptiStruct converges after 41 iterations, taking nearly 93.65 hours. This invention converges after 65 and 58 iterations at the two resolutions, respectively, with a total iteration time of 1.15 hours and 1.88 hours. This takes into account the preparation process performed before the optimization iteration begins (calculating the filtering matrix). Element stiffness matrix The time taken for the other two methods was 0.75h and 1.11h. Therefore, the actual optimization time of this invention was 1.90h and 2.99h, respectively, which is nearly 50 times and 31 times more efficient than the total time of OptiStruct, fully demonstrating the efficiency advantage of this invention. Figure 10 (b) provides a detailed description of the resolution. The average optimization time per step of the machine learning model is as follows. Due to the reduction of degrees of freedom using the substructure method, the dimensionality of the equilibrium equations is significantly reduced, and solving the finite element equations is no longer the computational bottleneck restricting optimization efficiency, accounting for only 13.3% of the computational cost per single optimization step. At this point, the computational cost of the exit stiffness matrix becomes significant, accounting for 60% and becoming the main part of the computational cost of the optimization iteration step. In the future, parallel computing techniques or simply assembling the upper triangular part of the stiffness matrix can be used to improve the computational efficiency of this part. Figure 10 (a) Time consumed in each optimization iteration step; (b) Based on A graph showing the percentage of average iteration time in resolution-based machine learning; The constraint function for the compliant mechanism problem is the same as that for the compliance minimization problem, only the objective function differs. The objective function for this problem is as follows: (20) in, To identify the indicator vector of the output displacement node; The compliance minimization optimization problem is relatively easy to solve due to its self-adjoint nature. In the second example, the generalization ability of the proposed machine learning model is verified through the non-self-adjoint problem of compliant mechanisms. Consider, for example... Figure 11 The compliant mechanism problem shown is optimized by taking one-quarter of the structure as the design domain based on symmetry. After optimization, the optimization results are mirror-symmetric to obtain the final topology. The design domain is divided into 400,000 fine elements, and a resolution of [resolution missing] is used. The machine learning model is optimized, with the density filtering radius set to three times the average size of the fine mesh. The volume fraction upper limit is set to 0.2. Objective function (OC) is used as the optimizer. The convergence condition is set to the condition that the relative change in the objective function value over the last five consecutive iterations is less than 1. The optimization process terminates at that time; Figure 12 The iterative history and intermediate configurations of the compliant structure problem are shown, with density contour plots indicating a threshold of 0.2. The optimization converged after 36 iterations, and the objective function... The structure was reanalyzed using the substructure method with linear interpolation of boundary displacements to obtain the baseline objective function value. The relative error is 1.3%. The optimal configuration has a hinge structure, which allows the structure to generate a reverse displacement response on the right side after being pushed on the left side, which meets the design expectation and verifies the effectiveness of the proposed machine learning model in solving non-self-adjoint problems, effectively demonstrating that the proposed model has good generalization ability. Figure 13 This demonstrates the compliant mechanism problem under different volume fraction constraints. The optimization results show a density cloud map with a threshold of 0.2. When the volume fraction is 0.1, the structural skeleton is relatively thin, and a hollow section appears on the left side of the structure, such as... Figure 13 As shown in (a), when the structure deforms under external force, the skeleton in this part is relatively thin, the deformation angle is large, and the strain energy is large, such as... Figure 13 As shown in (d). Therefore, the deformation transmitted to the right side is smaller. As the volume fraction increases to 0.2, the structural framework thickens, such as... Figure 13 As shown in (b), the left side of the structure has greater stiffness, and deformation energy is well transferred to the right side of the structure through hinges, as... Figure 13 As shown in (e). At this time, The amplitude is relatively large. When the volume fraction continues to increase to 0.3, The change from -2.29 to -2.31 is less than 1%. This is because as the volume fraction further increases, the excess material tends to be distributed on the right side of the hinge, i.e. Place, such as Figure 13 As shown in (c), this portion of the material does not significantly contribute to increasing the structure's ability to generate negative displacements. Its skeleton angle variation is similar to that of a volume fraction of 0.2, as... Figure 13 As shown in (f). The displacement contour plots of the structure under the three volume fractions are as follows. Figure 13 (gi) is shown. The corresponding iteration history is as follows: Figure 14 As shown. In all three cases, the objective function of the structure decreases smoothly and satisfies the volume constraint. Objective function Compared with the benchmark value The relative errors between them were 0.5%, 1.3%, and 1.7%, respectively, which effectively verified the generalization ability of the proposed machine learning model. Figure 13 (ac) Optimization results and magnified local details; (df) Configuration diagram (deformation magnification factor) ), and the objective function; (gi) displacement contour plot (deformation magnification factor) ); This application proposes a problem-independent transfer learning method for topology optimization of complex 3D design domain structures. It achieves efficient solutions to large-scale topology optimization problems under complex design domain conditions, expanding the application scope of problem-independent machine learning methods in complex geometric structures. By introducing transfer learning, it utilizes pre-trained models on regular substructures to predict the numerical shape functions of complex isoparametric substructures. Compared to direct training, it requires fewer training samples to achieve the same prediction accuracy, significantly reducing dataset construction and model training costs, and improving model training efficiency. Furthermore, it employs a neural network structure embedding mechanical information, introducing mechanical constraint information into the machine learning model, ensuring the completeness of the displacement field, and improving the prediction accuracy of substructure numerical shape functions and finite element analysis accuracy, thereby enhancing the reliability of topology optimization results. The machine learning model accelerates the finite element analysis process, significantly reducing computation time in topology optimization. Under serial computation conditions, compared to the 32-core parallel computation results of the commercial software OptiStruct, the overall optimization efficiency is still nearly 50 times higher, and the single-step optimization efficiency is improved by two orders of magnitude. Moreover, the established machine learning model is independent of the specific optimization problem, design domain, boundary conditions, and load conditions. When the optimization object or working condition changes, there is no need to readjust or retrain the model. It has strong versatility and generalization. It shows good accuracy and stability in compliance minimization, compliant mechanism optimization and topology optimization problems, and has good engineering application prospects. Since this invention aims to improve the efficiency of structural finite element analysis, its optimization process is consistent with general topology optimization methods. The complete calculation process is as follows: Figure 15 As shown.

[0024] Step 1. Discretization: Discretize according to a certain resolution ( or Discrete design domain to obtain substructure; Step 2: Preparation: Calculate the density filtering matrix Stiffness matrix of each fine element ; Step 3. Proceed to the optimization process. First, use the proposed method to predict the numerical shape function of each substructure. Next, the outlet stiffness matrix is ​​calculated; finally, the finite element equations are solved to obtain the substructure displacement response. ; Step 4. Use the obtained... and prediction Calculate the response of the fine mesh; Step 5. Sensitivity analysis; Step 6. Update the design variables using the optimizer OC (Optimality Criteria) or MMA (Method of Moving Asymptotes); Step 7. Convergence check. If the convergence criterion is not met, return to Step 3; otherwise, terminate the optimization process.

[0025] The purpose of this invention is to address the problems of high computational cost, high training cost of machine learning models, and insufficient generalization ability of existing structural topology optimization methods in complex design domains, and to propose an efficient structural topology optimization method for complex design domains using Problem-independent Transfer Learning (PITL).

[0026] This invention combines problem-independent machine learning methods with transfer learning, utilizing machine learning models pre-trained on regular substructures to achieve rapid prediction of numerical shape functions of complex isoparametric substructures. This reduces the dependence of machine learning models on a large number of training samples in complex design domains, lowers model training costs, and improves training efficiency. Furthermore, by constructing a neural network model embedding mechanical information, the prediction accuracy and stability of numerical shape functions of complex isoparametric substructures are improved, achieving efficient acceleration of the finite element analysis process in complex design domains. Moreover, while ensuring optimization accuracy, it significantly reduces the computation time of large-scale structural topology optimization in complex design domains, improving the application efficiency and engineering practicality of structural topology optimization methods in complex engineering problems.

[0027] The above description is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained in this invention are implemented according to conventional methods in the art unless otherwise specified or limited.

Claims

1. A method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning, characterized in that, Includes the following steps: S1: Generate samples for training machine learning models; S2: Construct and train the source task neural network for predicting the numerical shape functions of the rule substructure; S3: Based on the source task neural network, a target task neural network for predicting isoparametric substructure numerical shape functions is constructed and trained through transfer learning to obtain a problem-independent transfer learning model. S4: In the topology optimization process, the trained problem-independent transfer learning model is used to predict the numerical shape function of each substructure based on the material distribution and node coordinates. S5: Calculate the outlet stiffness matrix of each substructure based on the predicted numerical shape function, and assemble them to form the total stiffness matrix; S6: Update the design variables using the optimizer, iterate until convergence, and obtain the optimal topology.

2. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 1, characterized in that: The method for generating samples in S1 is the six-face intersection method, specifically including: Generate a point on the principal axis of the Cartesian coordinate system that is 1 distance from the origin. Randomly generate a vector as the normal vector. A plane is uniquely determined through this point. Repeat this process on all the positive and negative half axes of the coordinate system to obtain six planes. The planes intersect each other to form a hexahedron. The hexahedron is subdivided at a given resolution and filled with random material to obtain a substructure sample; finally, one node of the substructure is translated to the origin of the coordinate system, and the coordinates of all nodes are normalized.

3. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 2, characterized in that: The input to the source task neural network in S2 is the material distribution of the regular substructure and the coordinates of the minutiae, and the output is the numerical shape function corresponding to the minutiae; the source task neural network includes a first branch network and a backbone network; The calculation formula for the first branch network is: （1） The input layer of the first branch network is the first... Material distribution of regular substructures, i.e. This represents the density vector of the fine elements of the constituent substructure. This represents the activation function. These represent the first branch network, respectively. The weights, biases, and output values ​​of the hidden layers; the number of hidden layers in the first branch network is... .

4. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 3, characterized in that: The calculation formula for the backbone network in S2 is as follows: （2） （3） The input layer of the backbone network is the first... The first rule substructure The node coordinates of the detail point, i.e. ; coordinates of detail points All terms expanded using first, second, and third-degree complete polynomials are used as neurons in the actual input layer, i.e. ; These represent the backbone network's... The weights, biases, and output values ​​of the hidden layers, and the number of hidden layers in the backbone network are as follows: ; The output layer is obtained by performing a Hadamard product on the last hidden layers of the branch network and the backbone network. The weights and biases of the output layer are respectively used This indicates that, in the source task, the output layer is the first rule substructure. Numerical functions for detail points.

5. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 4, characterized in that: S3 specifically includes: Load the weights and biases of the already trained source task neural network and keep them unchanged; Based on the source task neural network, a second branch network is added. The input of this second branch network is the coordinates of the exit node of the isoparametric substructure. The calculation formula is as follows: （4） These represent the additional branch network's first branch, the second branch, and the third branch, the fourth branch, the fifth branch, and the sixth branch, ... seventh branch, and the sixth branch, respectively. The weights, biases, and output values ​​of the hidden layers, and the number of hidden layers are used. express; The last hidden layer of the second branch network The output layer of the original backbone network Perform a Hadamard product as the final output layer. And it is used to predict the numerical shape function of isoparametric substructures, that is: （5） During training, only the weights and biases of the second branch network are updated, and the loss function is the numerical shape function prediction. Compared with the true value The mean square error between them is calculated using the following formula: （6）。 6. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 5, characterized in that: In S4, when predicting numerical shape functions, the network output is constrained based on the invariance derived from mechanical principles to achieve the embedding of mechanical information.

7. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 6, characterized in that: The process of embedding the mechanical information is specifically as follows: Numerical shape functions of detail points inside the substructure Block by column and The rigid body displacement mode of the substructure exit node Block by line and ; Only As part of neural network prediction The following formula is used to calculate: （16） Among them, only As part of neural network prediction No longer output by a neural network, but based on Calculated, therefore, the stated Treat it as a predicted value, and use the above Treat it as a calculated value, derived from the predicted value. Compared with calculated values Numerical shape functions It satisfies invariance.

8. The method for efficient analysis and topology optimization of complex domains based on problem-independent transfer learning according to claim 1, characterized in that: The optimizer in step S6 is either the optimality criterion method or the moving asymptote method. Topology optimization problems are either compliance minimization problems or compliant mechanism problems. The optimization formula for the compliance minimization problem is as follows: （17） Among them, the design variable is the element density, which is determined by... This indicates that the total number of fine units is determined by... express, This represents the total number of substructures. If the substructure resolution is... Then there is , Let these represent the objective function and the constraint function, respectively. This represents the exit stiffness matrix of each substructure in the set. The obtained total stiffness matrix, These are the substructure displacement vector and load vector, respectively. For material volume, For the first The volume of fine units, This indicates the upper limit of the volume fraction.

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