A fast topology optimization method based on two-stage deep probabilistic model
By combining a two-stage deep probabilistic model with probabilistic intrinsic orthogonal decomposition and an encoder-decoder convolutional neural network, the problem of high computational cost in existing topology optimization is solved, and fast topology optimization on small sample datasets is achieved, which is applicable to two-dimensional and three-dimensional designs.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- 浣江实验室
- Filing Date
- 2023-11-22
- Publication Date
- 2026-07-24
AI Technical Summary
Existing deep probabilistic models are computationally expensive in topology optimization and are overly dependent on training datasets, making it difficult to achieve rapid design on small sample datasets.
A two-stage deep probabilistic model is adopted, which combines probabilistic intrinsic orthogonal decomposition and an encoder-decoder convolutional neural network. The training dataset is reduced by the linear assumption, and prediction is performed using prior information and Gaussian process. The encoder-decoder network enhances fine features.
Achieve fast topology optimization on small sample datasets, reduce computational costs, are applicable to 2D and 3D designs, shorten optimization time, and improve design accuracy.
Smart Images
Figure CN117540517B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the technical field of structural design, and in particular relates to a fast topology optimization method based on a two-stage deep probabilistic model. Background Technology
[0002] Topology optimization is a structural optimization method used to compute the distribution of shape or material within a defined domain. Mathematically, topology optimization can be formulated as a high-dimensional constrained optimization problem rooted in finite element analysis, where the objective function is to maximize the global performance of the design. Currently, the mainstream methods for topology optimization can be broadly classified into five categories: density-based methods, mesh-based methods, moving boundary-based methods, load path-based methods, and optimal partitioning methods.
[0003] Over the past few decades, topology optimization has been extensively studied and applied to various engineering problems, including static / dynamic mechanical structure design, eigenvalue problems, plastic deformation, design-related load studies, compliant mechanism design, and multiphysics problems. With recent advancements in additive manufacturing and related design solutions, topology optimization has demonstrated excellent performance in producing highly flexible, custom designs.
[0004] While topology optimization is a powerful design tool, certain processing is still required to obtain numerically stable results at an affordable computational cost. Deep probabilistic models have emerged that can predict near-optimal designs using a set of related conventional designs without simulation. Deep probabilistic models promise rapid design space exploration without explicitly setting up simulations. Furthermore, deep probabilistic model predictions can serve as initial guesses for further rapid topology optimization, such as in multiphysics or dynamic problems.
[0005] To improve the accuracy of deep probabilistic models, researchers have made significant efforts to incorporate the physical perspective of topology optimization into these models. Researchers at Arizona State University (R. Cang, H. Yao and Y. Ren, One-shot generation of near-optimal topology through theory-driven machine learning, Computer-Aided Design, 109 (2019) 12–21.) proposed an intrusive approach that uses simulation optimization criteria to iteratively add simulations from a random selection. Researchers at the Hong Kong University of Science and Technology (C. Qiu, S. Du and J. Yang, A deep learning approach for efficient topology optimization based on the element removal strategy, Materials & Design, 212(2021) 110179.) developed a convolutional recurrent neural network and predicted the spatiotemporal evolution of topology optimization. Researchers at MIT (AJ Lew and MJ Buehler, Encoding and exploring latent design space of optimal material structures via a VAE-LSTM model, Forces in Mechanics, 5 (2021) 100054.) combined a variant autoencoder with a long short-term memory neural network to predict the spatiotemporal evolution of topology optimization.
[0006] However, most existing deep probabilistic models are data-intensive, easily exceeding 10,000 training simulations, which is computationally challenging. Therefore, it is urgent to propose a deep probabilistic model for fast topology optimization that is suitable for small datasets and computationally efficient. Summary of the Invention
[0007] The purpose of this invention is to address the shortcomings of existing technologies by providing a fast topology optimization method based on a two-stage deep probabilistic model. This method reduces the dependence on the training dataset by imposing theoretical constraints as prior information, achieved by combining Probabilistic Eigenorthogonal Decomposition (POD) with an encoder-decoder convolutional neural network. POD assumes a linear relationship between the image dataset and the constantly changing boundary conditions. This linear assumption significantly reduces the size of the training dataset, avoiding heuristic optimization during model training. Due to this linear assumption, POD predictions are considered low-resolution; therefore, an encoder-decoder convolutional neural network is employed to enhance the finer features of the POD predictions.
[0008] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0009] A fast topology optimization method based on a two-stage depth probabilistic model includes the following steps:
[0010] (1) Determine the case for which topology design is to be carried out, and determine the loads, boundary conditions, objective functions and material volume constraints required in the generated design based on the actual working conditions of the case;
[0011] (2) The combination of loads and boundary conditions is sampled using Latin hypercube sampling to obtain prior constraints;
[0012] (3) Perform topology optimization on the cases based on prior constraints to construct the dataset;
[0013] (4) Construct a two-stage deep probability model and train the two-stage deep probability model on the constructed dataset;
[0014] (5) Use the trained two-stage deep probabilistic model to perform topology prediction on the case under unknown load and boundary conditions;
[0015] (6) The topology prediction results of the model are used as the basis for further topology optimization, thereby obtaining the optimal topology structure.
[0016] In the above technical solution, further, the case model for topology design described in step (1) is as follows:
[0017]
[0018] in the formula Let be the objective function. For density-related design variables, It is a displacement vector. Here is the stiffness matrix. Represents a force vector. Represents constraints;
[0019] For the compliance minimization problem , , This represents the volume of the structural portion containing materials. Represents the design domain volume. This represents the maximum permissible volume fraction threshold; each finite element is assigned a design variable, namely density, and density-related design variables. A value of 0 represents a void, and 1 represents a solid. The density variable is penalized by a power order, which determines the value of the responding physical quantity, such as material stiffness, cost, or electrical conductivity, as shown in the formula. As shown;
[0020]
[0021] In the design problem of minimizing flexibility It is the Young's modulus of a finite element. It is the Young's modulus of the given material; It is a density-related penalty coefficient that filters the density values of finite elements within a predefined radius to improve the numerical stability of topology optimization.
[0022] The selection of loads and boundary conditions is related to the actual working conditions. The number, magnitude, and direction of loads, as well as the number and range of boundary conditions, are selected to determine the loads and boundary conditions that need to be involved in the changes in this topology design case.
[0023] Furthermore, in step (2), the Latin hypercube sampling of the load and boundary conditions is performed according to the following rules:
[0024] After identifying the loads and boundary conditions that need to be changed, it is necessary to confirm the range of values for each variable. If the variable is a load, its quantity, magnitude, and direction can change. If the variable is a boundary condition, its quantity and coverage area can change. Let a be the total number of variables that need to be changed.
[0025] After confirming the variation range of each load and boundary condition, the total number of datasets b is constructed as needed. The sample space constructed by the range of a-dimensional variables is divided into b layers using Latin hypercube sampling. Then, random sampling is performed in each sample space and the order is shuffled to obtain the combination form in the variable space, thereby obtaining the prior constraints and ensuring the comprehensiveness of the sample results.
[0026] Furthermore, in step (3), the dataset is constructed according to the following rules:
[0027] Based on the results of Latin hypercube sampling, topology optimization is performed on cases with different combinations of variables. The obtained design variables and their corresponding positional relationships are saved. At the same time, the convergence curve of the objective function with the number of iterations in this process can be recorded for subsequent comparative analysis of computational costs.
[0028] The dataset obtained through topology optimization will be divided into training, validation, and test sets. The training and validation sets will be used for training the two-stage deep probabilistic model, while the test set will be used to verify the accuracy of the model's predictions.
[0029] Furthermore, the two-stage depth probability model described in step (4) is constructed according to the following rules:
[0030] The two-stage deep probabilistic model consists of two parts. The first part is a one-stage module, which uses the modes and their coefficients of probabilistic intrinsic orthogonal decomposition as the topology design of the first stage. The second part is a two-stage module, which uses an encoding-decoding convolutional neural network to perform nonlinear mapping on the topology design results of the one-stage module that lacks enhanced fine features.
[0031] In the first-stage module, probabilistic eigenorthogonal decomposition decomposes a given set of structural designs into a linear combination of fundamental modes. These fundamental modes are obtained through an optimizer, whose constraints are normalization and orthogonality. The topology design cases include two-dimensional and three-dimensional structures. For three-dimensional structures, the three-dimensional structure is transformed into a two-dimensional image. The pixel scalar values in the two-dimensional image format constitute a matrix. ,in Indicates the index of the image. ( = 1, 2, ..., ; = 1, 2, 3..., () represents the index of the pixel; each topology prediction All Basic modes and corresponding modal coefficients A linear combination, specifically as shown in the equation As shown;
[0032]
[0033] in, For modal indexing;
[0034] Fundamental modes and modal coefficients It is obtained by minimizing the root mean square error between the original image and the predicted value, as shown in the formula. As shown;
[0035]
[0036] formula and Normalization and orthogonality constraints were defined respectively;
[0037]
[0038]
[0039] Topology design adopts the previous Modality, where The number of varying loads and boundary condition variables; the probabilistic eigenorthogonal decomposition mode coefficients under unknown boundary conditions are approximated using a Gaussian process, which is well-suited for sparse data. ,in n is the i-th data point d 3D input vector, It is the relevant output ( = 1, 2, ..., n); Use the squared exponent check to design the Gaussian process, as shown in the formula. As shown;
[0040]
[0041] Where m represents the m-th dimension of the input. and Represents the process variance and length scale; the marginal likelihood of a Gaussian process requires an analysis of the covariance matrix. Perform inversion, covariance matrix The calculation is as follows: As shown:
[0042]
[0043] in It is an identity matrix; parameters , and We need to learn from the data; specifically, the parameters Controlling the length scale of the m-th dimension, while the parameter and Controlling variance and noise in a Gaussian process; given a dataset The boundary likelihood of a Gaussian process is as shown in the formula. As shown; in the first-stage module, the Metropolis-Hastings algorithm is used to estimate the posterior distribution of the parameters for fitting.
[0044]
[0045] The uncertainty of the first-stage topology design is estimated by the modes and their coefficients of the probabilistic eigenvalue decomposition; the pixel-by-pixel probabilistic eigenvalue decomposition prediction is essentially a band Gaussian distribution of 3D variables ,in The number of pixels in the image. Represents the load and boundary condition variables; average prediction results from probabilistic eigenorthogonal decomposition. As in the formula As shown, where These are the modal coefficients predicted by the Gaussian process;
[0046]
[0047] Assuming that the probabilistic eigenorthogonal decomposition modes are independent, then the covariance... It is a diagonal matrix, and its diagonal elements Pixel uncertainty in topology design The calculation formula is as follows:
[0048]
[0049] In the two-stage module, the encoder-decoder architecture convolutional neural network excels in image reconstruction by progressively capturing key features at different scales. The inputs to the encoder-decoder convolutional neural network include the load and boundary condition variables, the probability intrinsic orthogonal decomposition mean prediction result, and the probability intrinsic orthogonal decomposition prediction uncertainty. The encoding layer, latent space transformation layer, and decoding layer all employ a two-block structure. The ReLU layer is used for nonlinear mapping, and the sigmoid layer outputs pixel segmentation probabilities. The depth probabilistic model is formulated as a binary image segmentation problem to indicate the presence or absence of material in each pixel. The two-stage depth probabilistic model uses iterative fitting with maximum cross-entropy loss, which is calculated according to the formula... calculate;
[0050]
[0051] For single training data exist One-hot encoded label of a pixel, For single-encoding index, The maximum predicted value of the convolutional neural network is used; in the two-stage module, the existence of gray areas is eliminated by segmentation, the design boundary is better described, and the Adam optimizer is used for training optimization.
[0052] Furthermore, the topology design using the trained model described in step (5) is carried out according to the following rules:
[0053] In the constructed test set, unknown loads and boundary conditions outside the training and validation sets are sampled and selected. The specific values are input into the two-stage deep probabilistic model to obtain the corresponding topology prediction results. These results are then compared with the case structures obtained by using the gradient topology optimization method in the test set and the structures of other deep probabilistic models to verify the model's computational efficiency after training with prior constraints.
[0054] Furthermore, the model predictions shown in step (6) serve as the basis for subsequent topology optimization according to the following rules:
[0055] The predicted structure of the model is exported, and the correspondence between the design variables in the corresponding design domain is imported into the gradient topology optimization algorithm for subsequent topology optimization. The changes of the objective function with the number of iterations during the optimization process are recorded. The results obtained by this invention are compared with the results obtained by using the gradient topology optimization algorithm alone to obtain the changes in the overall computational cost.
[0056] The beneficial effects of this invention are as follows:
[0057] This invention is applicable to both 2D and 3D design, significantly reducing optimization time while maintaining design accuracy, and is not limited by design dimensions. By introducing probabilistic intrinsic orthogonal decomposition, this invention models the uncertainty of modal coefficients and propagates it to the topology design, serving as the basis for the first-stage low-fidelity topology. This demonstrates the feasibility of rapid topology design based on 100-200 training data points and 2-4 varying boundary condition variables. Through the development of an encoder-decoder convolutional neural network, the mean and uncertainty predicted by the first-stage synthetic design are used as the basis for nonlinear enhancement of fine features. This invention uses the results predicted by the two-stage deep probabilistic model as the basis for subsequent topology optimization, thereby significantly reducing the computational cost of topology optimization. Attached Figure Description
[0058] Figure 1 This is a flowchart of a fast topology optimization method based on a two-stage depth probability model;
[0059] Figure 2 The proposed encoding-decoding convolutional neural network structure diagram is shown below.
[0060] Figure 3A schematic diagram of loads and boundary conditions for a 3D machine tool column example;
[0061] Figure 4 Comparison of model prediction results with test set for the case of 3D machine tool column;
[0062] Figure 5 A comparison of the calculation results of this invention and a single gradient topology optimization algorithm is presented for a case study of a three-dimensional machine tool column.
[0063] Figure 6 A comparison of the computational costs of this invention and a standalone gradient topology optimization algorithm is presented for a three-dimensional machine tool column example. Detailed Implementation
[0064] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numerals in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention. Rather, they are merely examples of apparatuses and methods consistent with some aspects of the invention as detailed in the appended claims.
[0065] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The singular forms “a,” “the,” and “the” used in this invention and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more of the associated listed items.
[0066] It should be understood that although the terms first, second, third, etc., may be used in this invention to describe various information, this information should not be limited to these terms. These terms are only used to distinguish information of the same type from one another. For example, first information may also be referred to as second information without departing from the scope of this invention, and similarly, second information may also be referred to as first information. Depending on the context, the word "if" as used herein may be interpreted as "when," "when," or "in response to a determination."
[0067] The specific embodiments of the present invention will now be described more completely and clearly with reference to the accompanying drawings. Figure 1 A fast topology optimization method based on a two-stage depth probabilistic model includes the following steps:
[0068] (1) Determine the case for which topology design is to be performed, and determine the loads, boundary conditions, objective functions, and material volume constraints required for generating the design based on the actual working conditions of the case; specifically:
[0069] like Figure 3 As shown, the selected 3D machine tool column case is 500mm long, 500mm wide, and 1250mm high, corresponding to the initial design domain of the column. The spindle box installation corresponds to the upper working condition position, and the connection part is simplified to the upper working condition force surface. At this point, the objective function of the case is to minimize the structural flexibility, and the objective function and basic constraints are set as follows:
[0070]
[0071] in the formula For density-related design variables, It is a displacement vector. Here is the stiffness matrix. Represents a force vector. Represents the constraints; for the compliance minimization problem, , , This represents the volume of the structural portion containing materials. Represents the design domain volume. This represents the maximum allowed volume fraction threshold, which is set to 0.5 here;
[0072] Each finite element is assigned a design variable, density, where 0 represents voids and 1 represents solids. The density variable is penalized by a power order, which determines the value of the Young's modulus, a response physical quantity, as shown in the formula. As shown;
[0073]
[0074] In the design problem of minimizing flexibility, It is the Young's modulus of a finite element. It is the Young's modulus of the given material; This is a density-related penalty coefficient used to filter the density values of finite element samples within a predefined radius to improve the numerical stability of topology optimization; the material of the column solid structure model is set to cast iron. The density of cast iron... Poisson's ratio elastic modulus .
[0075] The selection of loads and boundary conditions is related to the actual working conditions. The number, magnitude, and direction of loads, as well as the number and range of boundary conditions, are selected to determine the loads and boundary conditions that need to be involved in the changes in this design case.
[0076] The column's operating conditions are set as follows: the bottom constraint surface and the upper working surface are subjected to corresponding forces and bending moments. The spindle box is installed on the upper working surface of the column. With the tool direction at the end of the spindle box selected as vertically downward, the tool experiences a force obliquely upward from the workpiece. In this example, the angle between the force and the vertical direction is selected as 45 degrees. The force on the spindle box tool is equivalently applied to the upper working surface of the column, resulting in a force at a 45-degree angle to the vertical direction and an equivalent bending moment. The equivalent force is defined as a concentrated load applied to the center point of the upper working surface, while the equivalent bending moment is applied to the entire upper working surface.
[0077] (2) The combination of loads and boundary conditions is sampled using Latin hypercube sampling; specifically:
[0078] After confirming the loads and boundary conditions that need to be varied, the range of values for the three variables needs to be determined. If the variable is a load, its quantity, magnitude, and direction can change; if the variable is a boundary condition, its quantity and coverage area can change. For this case, three variable load cases are designed, i.e., a=3. The first variable load case is the area of the bottom constraint surface. Keeping the length of the constraint surface constant, the width L of the constraint surface is changed, thus changing the constraint area. The value range is [300, 500], in mm. The second variable load case is the magnitude F of the concentrated load, with a value range of [10, 60], in kN. The third variable load case is the magnitude M of the bending moment, with a value range of [10, 60], in kN·m. The variable load case distribution is as follows: Figure 3 As shown;
[0079] After confirming the variation range of each load and boundary condition, the total number of datasets to be constructed was determined to be 500. Latin hypercube sampling was used to divide the sample space constructed by the 3D variable range into 500 layers. Then, random sampling was performed in each sample space and the order was shuffled to obtain the combination form in the variable space, ensuring the comprehensiveness of the sample results.
[0080] (3) Perform topology optimization on the cases based on prior constraints to construct a dataset; specifically:
[0081] Based on 500 sets of results from Latin hypercube sampling, topology optimization was performed on cases with different combinations of variables using the Method of Moving Asymptotes (MMA). The design variables in the obtained design domain and their corresponding positional relationships were saved. At the same time, the convergence curve of the objective function with the number of iterations in this process was recorded for subsequent comparative analysis of computational costs.
[0082] The 500 datasets obtained using MMA will be divided into training, validation, and test sets in a ratio of 8:1:1. The training and validation sets will be used for training the two-stage deep probabilistic model, while the test set will be used to verify the accuracy of the model's predictions.
[0083] (4) Construct a two-stage deep probabilistic model and train the two-stage deep probabilistic model based on the constructed dataset; specifically:
[0084] The two-stage deep probabilistic model consists of two parts. The first part is a one-stage module, which uses the modes and their coefficients from probabilistic eigenorthogonal decomposition as the topology design for the first stage. The second part is a two-stage module, which uses an encoder-decoder convolutional neural network to nonlinearly map the topology design result of the one-stage module, which lacks fine features. The shape of the initial input of the training data will be reset to... Voxel points of a dimension;
[0085] In the first-stage module, probabilistic eigenorthogonal decomposition decomposes a given set of structural designs into a linear combination of fundamental modes. These fundamental modes are obtained through an optimizer, whose constraints are normalization and orthogonality. The three-dimensional voxel input is transformed into a two-dimensional image, with the pixel scalar values in the image format forming a matrix. ,in Indicates the index of the image. ( = 1, 2, ..., ; = 1, 2, 3..., ) represents the index of the pixel; each topology prediction All Basic modes and corresponding modal coefficients Linear combinations, such as equation As shown, where Modal index;
[0086]
[0087] Fundamental modes and modal coefficients It is obtained by minimizing the root mean square error between the image and the predicted value, as shown in the formula. As shown;
[0088]
[0089] formula and Normalization and orthogonality constraints were defined respectively;
[0090]
[0091]
[0092] Topology design adopts the previous Modality, where The number of varying loads and boundary condition variables; the probabilistic eigenorthogonal decomposition mode coefficients under unknown boundary conditions are approximated using a Gaussian process, which is well-suited for sparse data. ,in n is the i-th data point d 3D input vector, It is the relevant output ( = 1, 2, ..., n); Use the squared exponent check to design the Gaussian process, as shown in the formula. As shown;
[0093]
[0094] Where m represents the m-th dimension of the input. and Represents the process variance and length scale; the marginal likelihood of a Gaussian process requires an analysis of the covariance matrix. Perform inversion, covariance matrix The calculation is as follows: As shown, where It is an identity matrix;
[0095]
[0096] In a Gaussian process, the parameters , and We need to learn from the data; specifically, the parameters Controlling the length scale of the m-th dimension, while the parameter and Controlling variance and noise in a Gaussian process; given a dataset The boundary likelihood of a Gaussian process is as shown in the formula. As shown; in the first-stage module, the Metropolis-Hastings algorithm is used to estimate the posterior distribution of the parameters for model fitting;
[0097]
[0098] The uncertainty of the first-stage module topology design is predicted by the modes and their coefficients of the probabilistic eigenorthogonal decomposition; the pixel-by-pixel probabilistic eigenorthogonal decomposition prediction is essentially a band Gaussian distribution of 3D variables ,in The number of pixels in the image. Represents the load and boundary condition variables; probability intrinsic orthogonal decomposition mean prediction results. As in the formula As shown, where These are the modal coefficients predicted by the Gaussian process;
[0099]
[0100] Assuming that the probabilistic eigenorthogonal decomposition modes are independent, then the covariance... It is a diagonal matrix, and its diagonal elements Pixel uncertainty in topology design The calculation formula is as follows:
[0101]
[0102] In the two-stage module, the encoder-decoder convolutional neural network performs excellently in image reconstruction by progressively capturing key features at different scales. The inputs to the encoder-decoder convolutional neural network include the load and boundary condition variables, the probability intrinsic orthogonal decomposition mean prediction result, and the probability intrinsic orthogonal decomposition prediction uncertainty. The basic network framework of the two-stage module is as follows: Figure 2 As shown; the encoding layer, latent space transformation layer, and decoding layer all adopt a two-block structure; the ReLU layer is used for nonlinear mapping, and the sigmoid layer is used to output pixel segmentation probabilities; the depth probability model is expressed as a binary image segmentation problem to indicate whether each pixel contains material; the two-stage depth probability model uses maximum cross-entropy loss for iterative fitting, and the maximum cross-entropy loss is based on the formula... calculate;
[0103]
[0104] For single training data exist One-hot encoded label of a pixel, For single-encoding index, The maximum predicted value of the convolutional neural network is given; in the two-stage module, a segmentation method is used to eliminate the existence of gray areas, better describe the design boundary, and the Adam optimizer is used for training optimization.
[0105] (5) Use the trained two-stage deep probabilistic model to perform topology design for the case under unknown load and boundary conditions; specifically:
[0106] In the remaining 50 test cases, unknown loads and boundary conditions outside the training and validation sets were sampled. Specific numerical values were input into the two-stage deep probabilistic model to obtain corresponding prediction results. These results were then compared with case structures obtained using gradient topology optimization in the test cases and structures from other deep probabilistic models to verify the model's computational efficiency after training with prior constraints. Four sets of cases with distinct shapes were selected for comparison, such as... Figure 4 As shown.
[0107] (6) Use the structure predicted by the model as the basis for further topology optimization, and compare the changes in overall computational cost; specifically:
[0108] The structures predicted by the deep probabilistic model for four selected cases under different loads and boundary conditions were exported. The correspondence between the design variables in the corresponding design domain was imported into the MMA algorithm for further topology optimization. The changes in the objective function with the number of iterations were recorded, and the results obtained by this invention were compared with the optimization results using only the MMA algorithm. Figure 5 As shown; then, 10 cases with different combinations of loads and boundary conditions were randomly selected from the test set, and the computational speed of using MMA alone for topology optimization and using deep probabilistic model prediction as the basis for subsequent topology optimization was analyzed and compared to obtain the change in overall computational cost (e.g. Figure 6 ).
[0109] The above embodiments are only used to illustrate the design concept and features of the present invention, and their purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes or modifications made based on the principles and design ideas disclosed in the present invention are within the protection scope of the present invention.
Claims
1. A fast topology optimization method based on a two-stage depth probabilistic model, characterized in that, Specifically, the following steps are included: (1) Determine the case for which topology design is to be carried out, and determine the loads, boundary conditions, objective functions and material volume constraints required in the topology design based on the actual working conditions of the case; (2) The combination of loads and boundary conditions is sampled using Latin hypercube sampling to obtain prior constraints; (3) Perform topology optimization on the cases based on prior constraints to construct the dataset; (4) Construct a two-stage deep probability model and train the two-stage deep probability model on the constructed dataset; (5) Use the trained two-stage deep probabilistic model to perform topology prediction on the case under unknown load and boundary conditions; (6) Use the topology prediction results of the model as the basis for further topology optimization, thereby obtaining the optimal topology structure. The two-stage depth probability model described in step (4) is constructed according to the following rules: The two-stage deep probabilistic model consists of two parts. The first part is a one-stage module, which uses the modes and their coefficients of probabilistic intrinsic orthogonal decomposition as the topology design of the first stage. The second part is a two-stage module, which uses an encoder-decoder convolutional neural network to perform nonlinear mapping on the topology design results of the one-stage module that lacks enhanced fine features. In the first-stage module, probabilistic eigenorthogonal decomposition decomposes a given set of structural designs into a linear combination of fundamental modes. These fundamental modes are obtained through an optimizer whose constraints are normalization and orthogonality. The pixel scalar values in the image format constitute a matrix. ,in Indicates the index of the image. The index representing the pixel, where = 1, 2, ... , ; = 1, 2, 3..., Each topology prediction All Basic modes and corresponding modal coefficients Linear combinations, such as equation As shown; （1） in, For modal indexing; Fundamental modes and modal coefficients It is obtained by minimizing the root mean square error between the original image and the predicted value, as shown in formula (2): （2） The constraints for normalization and orthogonality are defined as shown in formulas (3) and (4): （3） （4） Topology design adopts the previous Modality, where The number of varying loads and boundary condition variables; the probabilistic eigenorthogonal decomposition modal coefficients under unknown boundary conditions are approximated using a Gaussian process, specifically as follows: For sparse data ,in n is the i-th data point d 3D input vector, This is the relevant output. = 1,2, ..., n, the Gaussian process is designed using the square exponent check, as shown in formula (5): （5） in and Represents the process variance and length scale; m represents the m-th dimension of the input; covariance matrix The calculation is shown in formula (6): （6） Among them, parameters , and It needs to learn from the dataset. It is an identity matrix; parameters Controlling noise in a Gaussian process; given a dataset The boundary likelihood of the Gaussian process is shown in equation (7): （7） The uncertainty of the first-stage module topology design is predicted by the modes and their coefficients of the probabilistic eigenorthogonal decomposition; the pixel-by-pixel probabilistic eigenorthogonal decomposition prediction is essentially a band Gaussian distribution of 3D variables ,in The number of pixels in the image. Represents the load and boundary condition variables; probability intrinsic orthogonal decomposition mean prediction results. As shown in formula (8), where These are the modal coefficients predicted by the Gaussian process; （8） Assuming that the probabilistic eigenorthogonal decomposition modes are independent, then the covariance... It is a diagonal matrix, and its diagonal elements Pixel uncertainty in topology design The calculation formula is as follows: （9） In the two-stage module, the two-stage module sequentially includes an encoding layer, a latent space transformation layer, and a decoding layer, each layer adopting a two-block structure; the input of the encoding-decoding convolutional neural network includes the load and boundary condition variables, the probability intrinsic orthogonal decomposition mean prediction result, and the probability intrinsic orthogonal decomposition prediction uncertainty; The two-stage depth probability model is iteratively fitted using the maximum cross-entropy loss, which is calculated using formula (10): （10） For single training data exist One-hot encoded label of a pixel, For single-encoding index, This represents the maximum predicted value of the convolutional neural network; the Adam optimizer is used for training and optimization.
2. The fast topology optimization method based on a two-stage depth probabilistic model according to claim 1, characterized in that, The case model for topology design mentioned in step (1) is as follows: （11） in the formula Let be the objective function. For density-related design variables, It is a displacement vector. Here is the stiffness matrix. Represents a force vector. Represents constraints; Density-related design variables A value of 0 represents a hole, and 1 represents a solid; this is a density-related design variable. The physical quantities of the corresponding finite element response The calculation method is shown in formula (12); （12） in, It is the physical quantity of the response of a finite element. It is the physical quantity of the response of a given material; It is a density-related penalty coefficient; The selection of loads and boundary conditions is related to the actual working conditions. The number, magnitude, and direction of loads, as well as the number and range of boundary conditions, are selected to determine the loads and boundary conditions that need to be involved in the changes in this topology design case.
3. The fast topology optimization method based on a two-stage depth probabilistic model according to claim 1, characterized in that, In step (2), the Latin hypercube sampling of the load and boundary conditions is performed according to the following rules: If the variable is a load, what changes is its quantity, magnitude, and direction; if the variable is a boundary condition, what changes is its quantity and coverage; let a denote the total number of variables that need to participate in the change. To construct the total number b of the dataset as needed, the sample space constructed from the range of a-dimensional variables is divided into b layers using Latin hypercube sampling. Then, random sampling is performed in each sample space and the order is shuffled to obtain the combination form in the variable space, thus obtaining the prior constraints.
4. The fast topology optimization method based on a two-stage depth probabilistic model according to claim 1, characterized in that, The dataset described in step (3) is constructed according to the following rules: Based on the results of Latin hypercube sampling, topology optimization is performed on cases with different combinations of variables, and the resulting design variables and their corresponding positional relationships are saved. The dataset obtained through topology optimization will be divided into training, validation, and test sets. The training and validation sets will be used for training the two-stage deep probabilistic model, while the test set will be used to verify the accuracy of the model's predictions.
5. A fast topology optimization method based on a two-stage depth probabilistic model according to claim 1, characterized in that, Step (5) specifically involves: Unknown loads and boundary conditions outside the training and validation sets are sampled from the constructed test set, and the specific values are input into the two-stage deep probabilistic model to obtain the corresponding topology prediction results.