Multi-stage structure topological optimization method based on potential diffusion model

Abstract

The invention provides a multi-stage structure topological optimization method based on a potential diffusion model, and relates to the technical field of engineering design and structure design optimization. Creating a structure topology optimization image data set by using a traditional SIMP method and performing preprocessing; mapping the structure topological optimization image to a low-dimensional potential space for representation so as to reduce the data dimension and improve the calculation efficiency; a forward diffusion process and a reverse diffusion process are carried out on the initial potential variable in the low-dimensional potential pair, a denoising network is added in the reverse diffusion process, and a cross attention mechanism and a condition mechanism are introduced, so that a topological structure is adjusted and designed according to physical and functional requirements, and the structural fidelity is ensured; after training is completed, potential variables of a low-dimensional space are mapped back to a high-dimensional space, a topological structure is generated, few-step SIMP direct optimization is carried out on the topological structure, the structure is rapidly refined, physical information is directly injected into the design so as to adapt to given boundary conditions, and generation of the optimal topological structure can be completed through few-step iteration.

Description

Technical Field

[0001] The present invention relates to the technical field of engineering design and structural design optimization, and in particular to a multi-stage structural topology optimization method based on a potential diffusion model. Background Art

[0002] As an efficient engineering design method, topology optimization aims to maximize performance indicators by optimizing the distribution of materials under given constraints. This technology is widely used in aerospace, automotive, construction, biomedicine and other fields to design lightweight and high-performance structural parts.

[0003] At present, topology optimization methods are mainly divided into traditional optimization methods and deep learning-based methods. Traditional topology optimization methods, such as Solid Isotropic Material with Penalization (SIMP), require a large number of finite element iterations. These methods usually maximize structural stiffness or minimize flexibility by gradually adjusting the material distribution. However, these traditional methods have the problem of high computational resource consumption and easy to fall into local minima when dealing with large-scale or complex structures.

[0004] In recent years, the application of generative models based on deep learning in topology optimization has gradually attracted attention. Generative Adversarial Network (GAN) and Denoising Diffusion Probabilistic Model (DDPM) have been introduced into this field, showing potential advantages. For example, TopologyGAN improves the accuracy and adaptability of generated samples by introducing physical fields (such as Von Mises stress, strain energy density, and displacement field) as conditions. However, this method requires a large amount of Finite Element Analysis (FEA), which is computationally expensive, and the generated topology structure fails to fully take into account both manufacturability and performance goals. In addition, the GAN network is difficult to train and has limited generalization ability.

[0005] To overcome these problems, diffusion models have gradually become a new option. TopoDiff adopts a diffusion model and balances the needs of manufacturability and high-performance design by introducing physical fields as an additional guidance mechanism. Nevertheless, existing diffusion-based models still have problems such as slow iterative sampling process and high consumption of computing resources. In addition, these methods usually rely on a lot of preprocessing and proxy models to improve performance, which limits their practical applications.

[0006] Traditional methods and diffusion model-based methods have achieved certain results in topology optimization, but there are still problems such as low computational efficiency, high resource consumption, insufficient manufacturability of generated structures and insufficient performance optimization.

[0007] Traditional topology optimization methods, such as the solid isotropic material penalty method (SIMP), have a solid foundation in theory and application, but have the following defects:

[0008] 1. High consumption of computing resources: The SIMP method requires a large number of finite element iterations, and each iteration requires recalculating the stiffness matrix and stress distribution of the structure, resulting in a huge consumption of computing resources. This high amount of calculation makes the SIMP method inefficient when dealing with large-scale or complex structural problems, and it is difficult to meet the high efficiency requirements in actual engineering applications.

[0009] 2. Easy to fall into local minima: Since the SIMP method relies on the choice of initial design, the optimization process may fall into local minima rather than the global optimal solution. This limitation makes the optimization result may not be the optimal material distribution, limiting its application in complex design spaces.

[0010] Existing topology optimization methods based on deep generative models, such as generative adversarial networks (GANs) and diffusion-determined models (DDPMs), provide new ideas for topology optimization, but they also have obvious defects:

[0011] 1. Complex training and limited generalization ability: GAN models are prone to instability and mode collapse during training, requiring careful adjustment of parameters and training strategies. At the same time, GAN has limited generalization ability, and when dealing with unseen design conditions and complex structures, the generated results may not be accurate and practical enough.

[0012] 2. High computational cost: Methods such as TopologyGAN rely on a large amount of finite element analysis (FEA) for preprocessing and calculation of generation conditions. Each configuration of loads and boundary conditions requires running the FEA solver, resulting in high computational costs and making it difficult to be widely used in actual engineering.

[0013] 3. Slow iterative sampling process: Existing methods, such as TopoDiff, have made some progress in balancing manufacturability and performance design, but their iterative sampling process is slow and still requires a lot of computing resources. This high computing demand limits its efficiency in large-scale engineering applications.

[0014] 4. Strong dependence on preprocessing and surrogate models: These methods usually rely on a large number of preprocessing steps and surrogate models to improve performance. However, this dependence increases the computational complexity and implementation difficulty, making it difficult for the model to flexibly adapt to different design conditions and constraints in practical applications.

[0015] In summary, existing topology optimization methods, whether traditional SIMP methods or methods based on deep generative models, have defects such as large consumption of computing resources, complex training, limited generalization ability, high computational cost, and slow iteration process. These defects limit the wide application and efficiency of topology optimization technology in practical engineering. Therefore, there is an urgent need for an innovative topology optimization method that can improve computational efficiency, reduce resource consumption, and optimize the feasibility and manufacturability of the generated structure. Summary of the invention

[0016] In view of the problems of low computational efficiency, high resource consumption, insufficient manufacturability and performance optimization of the generated structure in existing topology optimization methods, the present invention proposes a multi-stage structural topology optimization method based on a potential diffusion model.

[0017] In order to solve the above technical problems, the technical solution adopted by the present invention is: a multi-stage structural topology optimization method based on a potential diffusion model, comprising the following steps:

[0018] Step 1: Determine the research object, create a structural topology optimization image dataset based on the traditional SIMP method and perform preprocessing;

[0019] Step 1.1: Select the classic rectangular beam as the research object;

[0020] Step 1.2: Establish a material interpolation model for design variables in the design domain based on the traditional SIMP method to penalize intermediate density values ​​and ensure the rationality and accuracy of material distribution;

[0021] The material interpolation model based on SIMP format is shown in the following formula:

[0022]

[0023] Among them, ρ e is the relative density of the elements in the design domain, E e (ρ e ) represents the elastic modulus of the interpolated unit, E 0 Represents the elastic modulus of the material of the solid part, E min is the elastic modulus of the hole part of the design domain, p is the penalty factor;

[0024] Step 1.3: Define the objective function and constraints for generating the structural topology optimization image dataset;

[0025] The objective function is to minimize the flexibility of the topological structure, as shown in the following formula:

[0026]

[0027] subject to:V(ρ) / V 0 =f

[0028] K(ρ)U=F

[0029] 0<ρ min ≤ρ≤1

[0030] Where C(ρ) is the flexibility value of the optimized structure, ρ is the density value of the design variable in the design domain, U and F are the displacement vector and load vector respectively, and k 0 is the initial element stiffness matrix, u e is the displacement vector of the unit, N is the number of units in the design domain, V 0 and V(ρ) are the initial volume and optimized volume of the design domain, respectively. f is the reserved fraction of the design domain volume. K(ρ) is the structural stiffness matrix of the design domain. min is the lower limit of the design variable density;

[0031] Step 1.4: Use random conditions to set physical parameters for the design domain, encode the design domain, obtain a structural topology optimization image dataset, and use data enhancement technology to expand the number of data samples for the structural topology optimization image dataset to obtain an expanded structural topology optimization image dataset;

[0032] Step 1.5: preprocessing the expanded structural topology optimization image data set to obtain a preprocessed structural topology optimization image data set;

[0033] Step 1.6: Divide the preprocessed structural topology optimization image dataset into a training set, a validation set, and a test set;

[0034] Step 2: Design an encoder network to map the multi-channel high-resolution structural topology optimization images in the structural topology optimization image dataset obtained in step 1 to a low-dimensional latent space representation, and obtain the initial latent variables corresponding to each multi-channel high-resolution structural topology optimization image;

[0035] Step 3: Perform a forward diffusion process on the initial latent variables obtained in step 2 in the low-dimensional latent space to obtain and store the noisy latent variables;

[0036] By gradually adding Gaussian noise, the encoded initial latent variables are transformed into pure noise variables, and the noisy latent variables generated at each step are stored for subsequent training of the reverse diffusion process;

[0037] Step 4: Train the inverse diffusion model based on the noisy latent variables, and optimize the denoising model parameters in the inverse diffusion model by minimizing the error between the real noise and the predicted noise;

[0038] Step 4.1: Design and train a denoising model to predict the noise component of each step in the reverse diffusion process and output the predicted value of the noise at each step;

[0039] Step 4.2: Establish a reverse diffusion target, minimize the error between the actual noise and the predicted value of the noise output by the denoising model through the loss function, and optimize the parameters of the denoising model;

[0040] Step 4.3: Use the loss function to update the denoising model parameters and train the reverse diffusion model;

[0041] Step 4.3.1: Input the noisy latent variables and the number of diffusion steps into the denoising model to obtain the predicted value of the output noise of the denoising model;

[0042] Step 4.3.2: Use the loss function to calculate the error between the real noise and the predicted value of the noise output by the denoising model, and propagate the gradient of the loss function from the output layer of the denoising model back to the input layer through the back-propagation technique, and adjust the denoising model parameters to minimize the loss;

[0043] Step 5: Use the denoising model to perform a reverse diffusion process on the pure noise variable, gradually remove the noise, and generate new latent variables until a new noise-free latent variable is generated;

[0044] The denoising model trained in step 4 is used to perform a reverse diffusion process on the pure noise variable obtained in step 3, and new potential variables are generated by gradually removing the noise. The reverse diffusion formula is:

[0045]

[0046] Where t is the number of diffusion steps, z t ' -1 is the new latent variable obtained when the number of reverse diffusion steps is t, z t ′ is the new latent variable obtained when the number of reverse diffusion steps is t+1, p θ (z t ' -1 |z t ′) represents the conditional probability distribution of the reverse process, that is, given z t ′, the latent variable z t ' -1 The probability distribution of μ θ (z t ′,t) and Σ θ (z t ′, t) are the predicted mean and predicted variance of the denoising model respectively;

[0047] At each step t, the predicted mean and predicted variance of the denoising model are used to generate a new latent variable z corresponding to the current diffusion step t by sampling from the conditional distribution t ' -1 , and its sampling formula is as follows:

[0048]

[0049] in, Represents the predicted mean μ of the denoising model θ (z t ′,t) and the prediction variance Σ θ (z t ′,t) normal distribution;

[0050] Step 6: Introduce the cross-attention mechanism and conditional mechanism in the reverse diffusion process to guide the reverse diffusion process;

[0051] Step 6.1: Encode the physical condition information through the pre-trained conditional encoder to obtain conditional embedding;

[0052] Step 6.2: Use the cross-attention mechanism to combine the conditional embedding with the new latent variables obtained in step 5 to guide the topology generation;

[0053] Step 6.3: Use the FiLM layer to introduce the conditional information into the normalization layer of the denoising model to enhance the role of the conditional information;

[0054] Step 7: Design a decoder network to decode the new noise-free latent variables obtained by the reverse diffusion process in step 5 and map them back to the pixel space, thereby generating a topological structure that is similar to the topological structure generated by the traditional SIMP method;

[0055] Step 8: Using the approximate topology generated in step 7 as a starting point, perform a few-step SIMP direct optimization to obtain the optimal topology;

[0056] Step 8.1: Use the approximate topology generated by the diffusion process as input to perform a few-step SIMP direct optimization;

[0057] Step 8.2: Set fixed boundary conditions and mechanical loads for the design domain as initial conditions for the few-step SIMP direct optimization;

[0058] Step 8.3: Use the finite element analysis method to perform mechanical analysis on the currently input topological structure and calculate the displacement and stress distribution of each design unit;

[0059] Step 8.4: According to the design update rule in the SIMP method, set the objective function of the few-step SIMP direct optimization, and adjust the material distribution by updating the design variables to maximize the stiffness of the topological structure under the volume or mass constraint, thereby minimizing the flexibility of the topological structure;

[0060] The objective function formula is as follows:

[0061]

[0062] in, The approximate topology generated in step 7 is is the flexibility value of the approximate topological structure, k e is the stiffness matrix of the design unit;

[0063] The design variable update formula is as follows:

[0064]

[0065] in, is the unit density at the kth iteration, η is the learning rate, is the gradient of the objective function with respect to density;

[0066] Step 8.5: Give a volume constraint to the optimization process so that the material volume of the topological structure does not exceed the volume of the design domain;

[0067] Step 8.6: Repeat the finite element analysis and design variable update until the preset number of iterations is reached, and output the final optimized topology structure.

[0068] The beneficial effect of adopting the above technical solution is that the present invention provides a multi-stage structural topology optimization method based on the potential diffusion model, which combines the potential diffusion model with traditional structural optimization technology. This innovation enables the topology optimization process to reduce data dimensions and computing requirements while enhancing the model's ability to handle complex designs. In addition, the introduction of the cross-attention mechanism in the present invention allows the potential diffusion model to adjust the design according to physical and functional requirements during the generation process, which is difficult to directly achieve in existing deep learning-based methods.

[0069] It is specifically reflected in the following aspects:

[0070] 1. Computational efficiency and speed are significantly improved: The diffusion process is transferred from the high-dimensional image space to the lower-dimensional latent space, which significantly reduces the dimension, memory usage and computational requirements of data processing, speeds up the generation of high-quality topological structures and reduces the consumption of computing resources.

[0071] 2. Fidelity and diversity of the generation process: By combining the diffusion model with U-Net denoising technology, the generated results can meet the design requirements and have high structural fidelity. By introducing the cross-attention mechanism and combining physical conditions, it is ensured that the generated topology not only performs well in performance but also meets manufacturing standards.

[0072] 3. Balance between performance and resource consumption: This paper integrates the traditional topology optimization method SIMP, uses a few-step direct optimization, uses the approximate optimal topology from the potential diffusion model as the starting point of optimization, quickly refines the structure and directly injects physical information into the design to adapt to given boundary conditions. This framework not only improves the diversity and efficiency of the structure, but also achieves a balance between performance and resource consumption.

[0073] 4. Wide applicability: The method proposed in the present invention is applicable to various resolutions and design requirements, and has good versatility and practical application value in different engineering fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 A flow chart of a multi-stage structural topology optimization method based on a potential diffusion model provided by an embodiment of the present invention;

[0075] Figure 2 A topological structure diagram generated at different resolutions and iteration steps by a multi-stage structural topology optimization method based on a potential diffusion model provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0076] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0077] In this embodiment, a multi-stage structural topology optimization method based on a potential diffusion model is proposed. This method maps a high-dimensional topology optimization problem into a low-dimensional potential space and uses a diffusion model to gradually generate an approximate optimal structure. In combination with the few-step SIMP optimization technology, the physical performance and manufacturing feasibility of the structure are further improved, thereby significantly reducing the computational cost while ensuring the design accuracy. Figure 1 As shown, the following steps are included:

[0078] Step 1: Determine the research object, create a structural topology optimization image dataset based on the traditional SIMP method and perform preprocessing;

[0079] Step 1.1: Select the classic rectangular beam as the research object;

[0080] Step 1.2: Establish a material interpolation model for design variables in the design domain based on the traditional SIMP method to penalize intermediate density values ​​and ensure the rationality and accuracy of material distribution;

[0081] The material interpolation model based on SIMP format is shown in the following formula:

[0082]

[0083] Among them, ρ e is the relative density of the unit in the design domain. When the value is 1, it means there is material, and when the value is 0, it means there is no material, that is, holes; E e (ρ e ) represents the elastic modulus of the design unit after interpolation, E 0 Represents the elastic modulus of the material of the solid part, E min is the elastic modulus of the hole part of the design domain to avoid the singularity of the stiffness matrix. p is the penalty factor. Usually p = 3 is selected as the penalty factor.

[0084] Step 1.3: Define the objective function and constraints for generating the structural topology optimization image dataset;

[0085] The objective function is to minimize the flexibility of the topological structure, as shown in the following formula:

[0086]

[0087] subject to:V(ρ) / V 0 =f

[0088] K(ρ)U=F

[0089] 0<ρ min ≤ρ≤1

[0090] Where C(ρ) is the flexibility value of the optimized structure, ρ is the density value of the design variable in the design domain, U and F are the displacement vector and load vector respectively, and k 0 is the initial element stiffness matrix, u e is the displacement vector of the unit, N is the number of units in the design domain, V 0 and V(ρ) are the initial volume and optimized volume of the design domain, respectively. f is the reserved fraction of the design domain volume. K(ρ) is the structural stiffness matrix of the design domain. min is the lower limit of the design variable density;

[0091] Step 1.4: After determining the research object and objective function, use random conditions to set different physical parameters including displacement boundaries, load conditions and volume fractions for the design domain, and encode the design domain. Through the random condition setting, the design domain is converted into a multi-channel high-resolution structural topology optimization image that can be processed by the neural network, and the corresponding structural topology optimization image dataset is generated. The number of data samples is expanded using data enhancement technology to obtain the expanded structural topology optimization image dataset. Where N is the number of image data samples, x i Represents a single image data sample, each image data sample x i It contains the density distribution of each unit in the design domain, that is, the density value of each unit in the design domain;

[0092] In this embodiment, the random conditions used when generating the data set include:

[0093] (1) The resolution of the rectangular beam design domain is set to 64×64 and 256×256;

[0094] (2) The volume fraction ranges from 0.2 to 0.8 with a step size of 0.02;

[0095] (3) The filter radius varies randomly from 1.5 to 6.0;

[0096] (4) The filtering method uses two methods: sensitivity filtering and density filtering;

[0097] (5) The number of forces applied to the subject ranged from 1 to 10, randomly distributed;

[0098] (6) Forces are randomly applied to any node in the design domain;

[0099] Data enhancement technology is an important means to improve the generalization ability of deep learning models, especially in the field of image processing. In this embodiment, the main data enhancement technologies used are as follows:

[0100] (1) Perform geometric transformations on topological image data, including rotation, translation, scaling, and flipping;

[0101] Among them, rotation is to randomly rotate the image by a certain angle, translation is to randomly move the image horizontally or vertically, scaling is to enlarge or reduce the image, and flipping is to flip the image horizontally or vertically;

[0102] (2) trimming and filling the topological structure image data, including random trimming and boundary filling;

[0103] Among them, random cropping is to randomly crop a region from the image, and border filling is to add background color or mirror extension to the edge of the image;

[0104] (3) Performing color transformation on topological structure image data, including brightness adjustment, contrast adjustment, saturation adjustment, and hue change;

[0105] Among them, brightness adjustment is to randomly change the brightness of the image, contrast adjustment is to adjust the contrast of the image, saturation adjustment is to randomly change the saturation of the image, and hue change is to adjust the hue of the image;

[0106] (4) adding noise to the topological structure image data, where the added noise types include Gaussian noise and salt and pepper noise;

[0107] Among them, Gaussian noise is to add random noise to the image, and salt and pepper noise is to randomly add black and white dots to the image.

[0108] Step 1.5: preprocessing the expanded structural topology optimization image data set obtained in step 1.4 to obtain a preprocessed structural topology optimization image data set;

[0109] The preprocessing method is to standardize the high-resolution structural topology optimization images in the data set and uniformly map the image pixel values ​​to the interval [0,1]. The standardization formula is as follows:

[0110]

[0111] Among them, x i High-resolution structural topology optimization images in the dataset, x i is the structural topology optimization image obtained after standardization, mean(x i ) and std(x i ) represent the mean and standard deviation of the image data samples respectively;

[0112] Step 1.6: In order to effectively train the neural network and accurately evaluate its performance, the preprocessed structural topology optimization image dataset obtained in step 1.5 is divided into a training set, a validation set, and a test set, with a division ratio of 8:1:1;

[0113] Step 2: Design an encoder network to map the multi-channel high-resolution structural topology optimization images in the structural topology optimization image dataset obtained in step 1 to a low-dimensional latent space representation, and obtain the initial latent variable z corresponding to each multi-channel high-resolution structural topology optimization image 0 ;

[0114] The encoder network converts high-dimensional multi-channel high-resolution topology optimization images into low-dimensional latent space representations by reducing the dimensionality of the input images, retaining the key features of the topological structure while significantly reducing the data dimension and processing complexity. Through this process, the most important information for optimization can be effectively extracted from the complex high-dimensional topological structure to provide support for subsequent optimization steps.

[0115] The encoder network structure is:

[0116] z 0 =E φ (x 0 )

[0117] Among them, x 0 Represents a single multi-channel high-resolution structural topology optimization image sample input to the encoder in the structural topology optimization image dataset X, E φ is the encoding function, z 0 represents the initial latent variable; φ is the learnable parameter of the encoder; where H and W represent the height and width of the initial multi-channel high-resolution structural topology optimization image data, respectively, and C represents the number of input channels;

[0118] Step 3: In the low-dimensional latent space, the initial latent variable z obtained in step 2 is 0 Implement the forward diffusion process to gradually add Gaussian noise to the encoded initial latent variable z 0 Transformed into pure noise variable z T , and the noisy latent variable z generated at each step t Storage, used for subsequent training of the reverse diffusion process;

[0119] The forward diffusion formula is:

[0120]

[0121] Among them, q(z t |z t-1 ) means that at a given z t-1 Under the condition of t The probability distribution of Indicates the mean and covariance is β t Normal distribution of Ι, z t is the current latent variable, z t-1 is the latent variable from the previous step, β t is the noise coefficient of the t-th step in the forward diffusion process, which indicates the proportion of Gaussian noise added in each step of the diffusion process, t = 1, 2, 3, ..., T, T is the number of diffusion steps, and Ι is the unit matrix;

[0122] In this embodiment, for each step t, the noisy latent variable z is generated according to the forward diffusion formula t , z t The calculation formula is expressed as:

[0123]

[0124] Among them, ∈ t represents noise, which follows a normal distribution with mean 0 and covariance as the identity matrix.

[0125] Step 4: Train the inverse diffusion model based on the noisy latent variables, and optimize the denoising model parameters in the inverse diffusion model by minimizing the error between the real noise and the predicted noise, so as to achieve efficient and accurate noise removal;

[0126] Step 4.1: Design a denoising model ∈ θ And train to predict the noise component of each step in the reverse diffusion process and output the predicted value of the noise at each step The input of the denoising model is the current step number t and the noisy latent variable z t , output the predicted value of noise at each step Used to approximate the actual noise in each step;

[0127] Step 4.2: Establish the reverse diffusion target and minimize the predicted value of the real noise ∈ and the noise output by the denoising model through the loss function The error between them is used to optimize the parameters of the denoising model;

[0128] The inverse diffusion model optimization process measures the difference between the noise predicted by the denoising model and the actual noise through a loss function. The loss function used in this embodiment is the mean square error:

[0129]

[0130] in, is the mean square error loss function, θ is the parameter of the denoising model, ∈ is the real noise sampled from the Gaussian distribution, ∈ θ (z t ,t) is the prediction of the denoising model for the noise.

[0131] Step 4.3: Use the loss function to update the denoising model parameters θ and train the reverse diffusion model;

[0132] Step 4.3.1: Substitute the noisy latent variable z t And the diffusion step number t are input into the denoising model to obtain the predicted value of the denoising model output for the noise

[0133] The prediction value formula of noise is:

[0134]

[0135] Among them, ∈ θ is a denoising model with parameter θ and its output is an estimate of the noise

[0136] Step 4.3.2: Use the mean squared error loss function Calculate the predicted value of the real noise ∈ and the noise output by the denoising model The error between them is calculated, and the gradient of the loss function is propagated from the output layer of the denoising model back to the input layer through the back-propagation technique, and the denoising model parameters θ are adjusted to minimize the loss.

[0137] Backpropagation is an important technology for optimizing machine learning models. It calculates the gradient of the loss function relative to the model parameters through the chain rule, thereby determining the adjustment direction and size of each parameter in the current step. Backpropagation can propagate error signals layer by layer and apply optimization algorithms such as gradient descent to update model parameters to better fit the training data.

[0138] Step 5: Use the denoising model to perform a reverse diffusion process on the pure noise variable, gradually remove the noise, and generate new latent variables until a new noise-free latent variable z′ is generated. 0 ;

[0139] Use the denoising model trained in step 4 to denoise the pure noise variable z obtained in step 3 T Perform a reverse diffusion process to generate new latent variables z by gradually removing noise t ′, the reverse diffusion formula is:

[0140]

[0141] Among them, p θ (z t ' -1 |z t ′) represents the conditional probability distribution of the reverse process, that is, given z t ′, the latent variable z t ' -1 The probability distribution of μ θ (z t ′,t) and Σ θ (z t ′,t) are the predicted mean and predicted variance of the denoising model, respectively, and are defined as follows:

[0142] The predicted mean μ of the denoising model θ (z t ′,t) formula is:

[0143]

[0144] The prediction variance of the denoising model Σ θ (z t ′,t) formula is:

[0145]

[0146] in, is the noise variance corresponding to time step t, which can usually be expressed as β t , which is the noise coefficient at each step in the forward diffusion process, and I is the unit matrix, which ensures that the variance is equal in each dimension;

[0147] At each step t, the predicted mean and variance of the denoising model are used to generate a new latent variable z corresponding to the current step t by sampling from the conditional distribution t ' -1 , and its sampling formula is as follows:

[0148]

[0149] in, Represents the predicted mean μ of the denoising model θ (z t ′,t) and the prediction variance Σ θ (z t ′,t) is a normal distribution.

[0150] Step 6: Introduce the cross-attention mechanism and conditional mechanism in the reverse diffusion process to guide the reverse diffusion process;

[0151] In the reverse diffusion process, physical condition information (such as displacement boundaries, load conditions, and volume fractions) is incorporated into the topology optimization generation process through the cross-attention mechanism and the condition mechanism. These two mechanisms ensure that the generated topology meets the design requirements (such as flexibility minimization, volume constraints, and boundary conditions), thereby accurately controlling the physical properties of the generated structure.

[0152] Step 6.1: Encode the physical condition information (including displacement boundary, load condition and volume fraction, etc.) through the pre-trained conditional encoder to obtain the conditional embedding τ θ ; This embedding contains key information about the topological structure in a specific physical environment, such as boundary constraints and force conditions;

[0153] Step 6.2: In the reverse diffusion process, use the cross-attention mechanism to embed the conditional embedding τ θ With the new latent variable z t ′ combined to guide the generation of topological structure;

[0154] Embedding the condition into τ θAs key K and value V, the new latent variable z t ′ is used as the query Q. By calculating the similarity between the query Q and the key K, the topological structure generation is guided to ensure that it meets the physical conditions;

[0155] The calculation formula of cross attention is as follows:

[0156]

[0157] Among them, d k is the dimension of the key, Q is the potential representation z t The query vector is K, K is the key vector of the condition information, and V is the value vector of the condition information.

[0158] Q, K, V can be expressed as follows:

[0159]

[0160] Among them, W Q , W K , W V are the linear transformation matrices of query, key, and value, respectively. Represents the denoising model ∈ θ The (flattened) intermediate feature representation of the U-Net network in .

[0161] Through the cross-attention mechanism, conditional information is fused with latent variables to control the influence of physical conditions on the geometric and mechanical characteristics during the generation process. The cross-attention ensures that the generated topological structure gradually meets the design goals such as minimization of flexibility and volume constraints.

[0162] Step 6.3: Use the FiLM layer to introduce the conditional information into the normalization layer of the denoising model to enhance the role of the conditional information;

[0163] The FiLM layer adjusts the features in the denoising model so that the conditional information flexibly affects the structure generation at each step. The calculation formula is as follows:

[0164] FiLM(h(z t ′))=γ(τ θ )·h(z t ′)+β(τ θ )

[0165] Among them, γ(τ θ ) and β(τ θ ) are the scaling and offset parameters generated by the conditional embedding, h(z t ′) represents the latent variable z with input noise t ′.

[0166] Through,FiLM layers, the characteristic response is further adjusted to ensure that the generated structure gradually reaches the optimization target.,The final generated structure not only meets the requirements of minimizing flexibility, but also complies with the given boundary conditions and volume constraints, ensuring its design performance and manufacturing feasibility.

[0167] In the reverse diffusion process, the latent variable z generated at each step t ′ are all connected through the cross attention mechanism and the physical condition τ θ The fusion is carried out to ensure that the generation process of the topological structure can gradually meet the design goals; the final generated structure not only meets the optimization goal of minimizing flexibility, but also conforms to the given boundary conditions and volume constraints, ensuring its design performance and manufacturing feasibility.

[0168] Step 7: Design the decoder network D ψ , the new noise-free latent variable z′ obtained by the reverse diffusion process in step 5 0 Decoding and mapping it back to pixel space results in a topology that is similar to that generated by the traditional SIMP method To ensure that the generated topology structure can retain the details in the design and has high fidelity; the decoder uses a deconvolutional neural network to reconstruct the topology structure.

[0169] The specific decoding process is as follows: Input the new latent variable z′ generated by reverse diffusion 0 , the decoder performs deconvolution operations, reconstructs the spatial structure of the latent variables layer by layer, and outputs a topological structure that is similar to the topological structure generated by the traditional SIMP method And to ensure that its details are accurately presented, the decoder network structure is:

[0170]

[0171] Among them, D ψ is the decoding function.

[0172] Step 8: Use the approximate topology generated in step 7 As a starting point, perform Few-step SIMP direct optimization (FS-SIMP) to obtain the optimal topology

[0173] The topology generated in step 7 It is visually identical to the original high-resolution optimal topology x 0 However, in order to ensure that the generated topology has the minimum flexibility under given boundary conditions and satisfies volume or mass constraints to ensure the efficiency and manufacturability of the structural design, a few-step SIMP iterative optimization is required.

[0174] Few-step SIMP direct optimization (FS-SIMP) is based on SIMP (Solid Isotropic Material with Penalization) topology optimization strategy to further optimize the generated approximate optimal topology. Through a small number of iterative steps, the material distribution is optimized, the structural stiffness is maximized under the premise of satisfying boundary conditions and volume constraints, and finally the optimal structural design is achieved.

[0175] Step 8.1: Utilize the approximate topology generated by the diffusion process As input, perform few-step SIMP direct optimization;

[0176] Step 8.2: Set fixed boundary conditions and mechanical loads for the design domain as the initial conditions for the few-step SIMP direct optimization. These boundary conditions define the physical constraints of the topology, such as constraint positions and loading methods.

[0177] Step 8.3: Use the finite element analysis method (FEA) to perform mechanical analysis on the currently input topological structure and calculate the displacement and stress distribution of each design unit;

[0178] The finite element equation is as follows:

[0179]

[0180] Among them, K is the global stiffness matrix, which is composed of the stiffness matrix k of each design unit e Composition, U is the displacement vector of the structure; F is the external load vector.

[0181] Step 8.4: According to the design update rule in the SIMP method, the objective function of the few-step SIMP direct optimization is set, and the material distribution is adjusted by updating the design variables (element density) so as to maximize the stiffness of the topological structure under the volume or mass constraint, thereby minimizing the flexibility of the topological structure (i.e., maximizing the stiffness);

[0182] The objective function formula is as follows:

[0183]

[0184] in, The approximate topology generated in step 7 is is the flexibility value of the approximate topological structure, k e is the stiffness matrix of the design unit;

[0185] The design variable update formula is as follows:

[0186]

[0187] in, is the unit density at the kth iteration, η is the learning rate (update step size), is the gradient of the objective function with respect to density;

[0188] Step 8.5: Give a volume constraint to the optimization process so that the material volume of the topological structure does not exceed the volume of the design domain;

[0189] The material volume calculation formula for the topological structure is:

[0190]

[0191] in, is the material volume of the topological structure, v e is the material volume of design unit e, V 0 is the volume of the design domain;

[0192] Step 8.6: Repeat the finite element analysis and design variable update until the preset number of iterations is reached and the final optimized topology is output.

[0193] This step not only significantly improves the performance and manufacturability of the generated topological structure, but also achieves a good balance between computational efficiency and optimization accuracy, fully demonstrating the great potential of topology optimization in practical applications.

[0194] In this embodiment, the multi-stage topology optimization method based on the potential diffusion model (TOLDM) proposed in the present invention uses the generated approximate optimal topology structure As a starting point, high-quality topologies were generated on datasets with 64×64 and 256×256 resolutions by direct optimization with few-step SIMP, e.g. Figure 2 As shown. Figure 2In the figure, the left and right sides are visualization results with different resolutions of 64×64 and 256×256 respectively. The first column of the visualization results with two resolutions shows the real benchmark result (GT) obtained using the traditional SIMP method. TOLDM(0), TOLDM(5) and TOLDM(10) respectively show the topological structures under different optimization steps. Among them, TOLDM(0) is the topological structure without SIMP method optimization iteration, TOLDM(5) is the result after 5 steps of SIMP method iterative optimization, and TOLDM(10) is the result after 10 steps of SIMP method iterative optimization. It can be intuitively observed from these visualization results that compared with the benchmark result (GT), the topological structures generated by TOLDM(5) and TOLDM(10) are getting closer and closer to GT in terms of boundary conditions and overall structure. This shows that the topological optimization method of the present invention performs well in terms of flexibility minimization and volume constraint objectives, and proves that TOLDM can generate high-quality topological optimization structures that meet the constraints, which fully verifies the effectiveness of the method proposed in the present invention.

[0195] In order to comprehensively evaluate the performance of the proposed method and compare it with similar works, this embodiment uses a series of evaluation indicators covering key requirements of physics, engineering and modeling. These indicators not only measure the overall performance of the proposed method, but also reflect its ability to generate feasible topological structures. The following is a detailed description of the key evaluation indicators:

[0196] 1. Compliance error (CE): evaluates the relative error between the TOLDM generated result and the traditional SIMP method. The calculation formula is:

[0197]

[0198] in, is the flexibility result of the TOLDM of the present invention, C(x 0 ) is the flexibility result of the traditional method SIMP. We also introduce the average flexibility error (CE%AVG) and median flexibility error (CE%MDN) to comprehensively evaluate the model performance.

[0199] 2. Volume Fraction Error (VFE): This indicator is used to evaluate the deviation between the volume fraction of the topological structure generated by the TOLDM algorithm and the actual input volume fraction. In order to more comprehensively evaluate the volume fraction accuracy of TOLDM, we also introduced the average volume fraction error (VFE%AVG) and the median volume fraction error (VFE%MDN). The calculation formula is:

[0200]

[0201] in, is the volume fraction result of TOLDM of the present invention, V(x 0 ) is the volume fraction structure of the traditional method SIMP.

[0202] 3. Inference Time: Measures the inference efficiency of the TOLDM model, including preprocessing and sampling time, focusing on evaluating its speed and efficiency in generating topology optimization structures.

[0203] In Tables 1 and 2, we show the performance evaluation results of TOLDM on 64×64 and 256×256 resolution datasets. Table 1 details the performance of each model on the 64×64 dataset. A comprehensive comparison is made with TopologyGAN, cDDPM, Consistency Model, TopoDiff, TopoDiffw / G, DOM w / TA and other methods. The results show that:

[0204] In contrast, TopologyGAN and cDDPM show higher average flexibility error (CE%AVG) values ​​of 48.51 and 60.79, respectively, which is mainly due to the convergence difficulties and lack of effective physical constraints in these models, resulting in large deviations in the manufacturability and practical applicability of the generated topology. It is worth noting that when the few-step SIMP direct optimization (FS-SIMP) step values ​​are 5 and 10, TOLDM shows the lowest values ​​in CE%AVG and CE%MDN values, which decrease from 4.23 to 2.46 and from 0.65 to 0.29, respectively, showing the smallest average error and median error in the topology optimization generation task, demonstrating excellent performance and stability. Similarly, the VFE%AVG and VFE%MDN values ​​of TOLDM also decrease with the increase of FS-SIMP, reaching 0.83 and 0.46, respectively, when the FS-SIMP step value is set to 10, which is the best performance among all models. In general, the TOLDM model with FS-SIMP step values ​​set to 5 and 10 showed consistent superiority in all indicators, highlighting its efficiency and robustness in topology optimization structure generation, and has significant advantages over other models.

[0205] Table 2 clearly shows that our TOLDM framework shows significant advantages on the dataset with 256×256 resolution. Even in higher resolution settings, TOLDM successfully achieves a balance between computational complexity and model performance. When the FS-SIMP step value of TOLDM is set to 5 and 10, the model maintains the lowest average volume fraction error (VFE%AVG) while achieving a very low average flexibility error (CE%AVG), which is significantly better than other models. After 5 steps of direct optimization, the performance of TOLDM is significantly better than other methods. In particular, when the FS-SIMP value is set to 10, the values ​​of the average flexibility error (CE%AVG) and the average volume fraction error (VFE%AVG) are reduced to 2.73 and 1.07, while the values ​​of the median flexibility error (CE%MDN) and the median volume fraction error (VFE%MDN) are also reduced to 0.39 and 0.72, respectively. The optimized topology of the present invention shows a significant leading advantage in these two key performance indicators, which shows the stability and reliability of the TOLDM method. These results not only demonstrate the stability and reliability of the TOLDM approach, but also reinforce its practicality as an optimization framework, proving that the model can achieve high-precision optimization from low to high complexity.

[0206] These quantitative analyses show that the TOLDM model performs well in multiple evaluation indicators, especially in terms of flexibility and volume fraction, where the errors are extremely small, further confirming the efficiency and robustness of the model in topology optimization tasks.

[0207] Overall, TOLDM achieves accurate and stable performance in both high and low model complexity settings. TOLDM shows significant performance improvements when the number of direct optimization iterations increases, whether on 64×64 or 256×256 resolution datasets. These results demonstrate that TOLDM can adapt to different resolutions and optimization requirements, providing a framework with strong adaptability and optimization accuracy in image generation and reconstruction tasks. TOLDM performs well on these metrics, showing significant advantages in reducing the mean and median errors, reflecting its superior ability to generate high-fidelity topology optimized structures. Compared with other models, the experimental results demonstrate the potential practicality and superiority of TOLDM in the field of topology optimization generation. In particular, when the number of FS-SIMP steps is 10, TOLDM outperforms other models in all metrics, which emphasizes its ability to significantly improve the topology structure with high performance through a small number of SIMP iterations.

[0208] One of the main indicators of data-driven design methods is reasoning time and generation efficiency. In Table 3, we compare the traditional SIMP method and the existing diffusion model-based methods in terms of reasoning time. The main goal of these deep learning solutions for data-driven design is to generate topology faster than traditional optimization. Table 3 mainly compares the average reasoning time of different models at low resolution 64×64 and high resolution 256×256. TOLDM includes 10 SIMP iterations when calculating the reasoning time. In the experimental results in Table 3, we can see that TOLDM is significantly faster than other state-of-the-art models at 64×64 and 256×256 resolutions. In particular, for 64×64 resolution, TOLDM's reasoning time is 96.63% faster than the classic SIMP method and 25.61% faster than DOM. It is 26% faster than DOM, the most advanced model for solving topology optimization problems based on diffusion models. This proves that our proposed method TOLDM can regenerate topological structures with both high efficiency and excellent performance. At the same time, in order to further demonstrate the superiority of our method, we analyze the test results of high resolution 256×256. According to the experimental results in Table 3, it can be clearly seen that TOLDM is still the most efficient compared with other methods at a resolution of 256×256, and its advantage is more obvious in the case of high resolution than in the case of low resolution, which further proves the efficiency and scalability of TOLDM.

[0209] The multi-stage topology optimization method based on latent diffusion model (TOLDM) proposed in the present invention is a new deep learning framework for topology optimization, which can effectively improve the accuracy and show good results in satisfying constraints, manufacturability and performance. TOLDM performs significantly better than similar diffusion model-based methods in multiple resolutions and fields.

[0210] Through the results of numerical experiments, it can be concluded that TOLDM significantly outperforms similar diffusion model-based methods in performance. This method can not only be more easily extended to higher-dimensional data, but also significantly improves computational efficiency and speed. In addition, it optimizes the feasibility and manufacturability of the generated structure, achieving an effective balance between performance and resource consumption. TOLDM also introduces physical constraints through the characteristics of its latent space and the cross-attention mechanism, making it easier to introduce novel, feasible, and manufacturing-satisfactory topological structures during the generation process.

[0211] These results demonstrate that the TOLDM model is not only capable of generating high-quality topological structures, but also has significant advantages in computational efficiency, especially at high resolutions, where both performance and speed are optimized.

[0212] Table 1 Quantitative evaluation results of the dataset with a resolution of 64×64

[0213]

[0214] Table 2 Quantitative evaluation results of the dataset with a resolution of 256×256

[0215]

[0216]

[0217] Table 3 Average inference time of different methods at different resolutions

[0218]

[0219] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A multi-stage structural topology optimization method based on a potential diffusion model, characterized in that: Step 1: Determine the research object, create a structural topology optimization image dataset based on the traditional SIMP method and perform preprocessing; Step 2: Design an encoder network to map the multi-channel high-resolution structural topology optimization images in the structural topology optimization image dataset obtained in step 1 to a low-dimensional latent space representation, and obtain the initial latent variables corresponding to each multi-channel high-resolution structural topology optimization image; Step 3: Perform a forward diffusion process on the initial latent variables obtained in step 2 in the low-dimensional latent space to obtain and store the noisy latent variables; Step 4: Train the inverse diffusion model based on the noisy latent variables, and optimize the denoising model parameters in the inverse diffusion model by minimizing the error between the real noise and the predicted noise; Step 5: Use the denoising model to perform a reverse diffusion process on the pure noise variable, gradually remove the noise, and generate new latent variables until a new noise-free latent variable is generated; Step 6: Introduce the cross-attention mechanism and conditional mechanism in the reverse diffusion process to guide the reverse diffusion process; Step 7: Design a decoder network to decode the new noise-free latent variables obtained by the reverse diffusion process in step 5 and map them back to the pixel space, thereby generating a topological structure that is similar to the topological structure generated by the traditional SIMP method; Step 8: Using the approximate topology generated in step 7 as a starting point, perform a few-step SIMP direct optimization to obtain the optimal topology.

2. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 1, characterized in that: The specific method of step 1 is: Step 1.1: Select the classic rectangular beam as the research object; Step 1.2: Establish a material interpolation model for design variables in the design domain based on the traditional SIMP method to penalize intermediate density values ​​and ensure the rationality and accuracy of material distribution; The material interpolation model based on SIMP format is shown in the following formula: Among them, ρ e is the relative density of the elements in the design domain, E e (ρ e ) represents the elastic modulus of the interpolated unit, E0 represents the elastic modulus of the solid part material, E min is the elastic modulus of the hole part of the design domain, p is the penalty factor; Step 1.3: Define the objective function and constraints for generating the structural topology optimization image dataset; The objective function is to minimize the flexibility of the topological structure, as shown in the following formula: subject to:V(ρ) / V0=f K(ρ)U=F 0<ρ min ≤ρ≤1 Where C(ρ) is the flexibility value of the optimized structure, ρ is the density value of the design variable in the design domain, U and F are the displacement vector and load vector respectively, k0 is the initial unit stiffness matrix, and u e is the displacement vector of the unit, N is the number of units in the design domain, V0 and V(ρ) are the initial volume and optimized volume of the design domain, f is the reserved fraction of the design domain volume, K(ρ) is the structural stiffness matrix of the design domain, ρ min is the lower limit of the design variable density; Step 1.4: Use random conditions to set physical parameters for the design domain, encode the design domain, obtain a structural topology optimization image dataset, and use data enhancement technology to expand the number of data samples for the structural topology optimization image dataset to obtain an expanded structural topology optimization image dataset; Step 1.5: preprocessing the expanded structural topology optimization image data set to obtain a preprocessed structural topology optimization image data set; Step 1.6: Divide the preprocessed structural topology optimization image dataset into training set, validation set and test set.

3. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 1, characterized in that: The step 3 converts the encoded initial latent variable into a pure noise variable by gradually adding Gaussian noise, and stores the noisy latent variable generated in each step for subsequent training of the reverse diffusion process.

4. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 1, characterized in that: The specific method of step 4 is: Step 4.1: Design and train a denoising model to predict the noise component of each step in the reverse diffusion process and output the predicted value of the noise at each step; Step 4.2: Establish a reverse diffusion target, minimize the error between the actual noise and the predicted value of the noise output by the denoising model through the loss function, and optimize the parameters of the denoising model; Step 4.3: Use the loss function to update the denoising model parameters and train the inverse diffusion model.

5. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 4, characterized in that: The specific method of step 4.3 is: Step 4.3.1: Input the noisy latent variables and the number of diffusion steps into the denoising model to obtain the predicted value of the output noise of the denoising model; Step 4.3.2: Use the loss function to calculate the error between the true noise and the predicted value of the noise output by the denoising model, and through the back-propagation technique, propagate the gradient of the loss function from the output layer of the denoising model back to the input layer, and adjust the denoising model parameters to minimize the loss.

6. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 1, characterized in that: The step 5 uses the denoising model trained in step 4 to perform a reverse diffusion process on the pure noise variable obtained in step 3, and generates a new potential variable by gradually removing the noise. The reverse diffusion formula is: Where t is the number of diffusion steps, z′ t-1 is the new latent variable obtained when the number of reverse diffusion steps is t, z′ t is the new latent variable obtained when the reverse diffusion step is t+1, p θ (z′ t-1 |z′ t ) represents the conditional probability distribution of the reverse process, that is, given z′ t Under the condition of t-1 The probability distribution of μ θ (z′ t ,t) and Σ θ (z′ t ,t) are the predicted mean and predicted variance of the denoising model respectively; At each step t, the predicted mean and predicted variance of the denoising model are used to generate a new latent variable z′ corresponding to the current diffusion step t by sampling from the conditional distribution t-1 , and its sampling formula is as follows: in, Represents the predicted mean μ of the denoising model θ (z′ t ,t) and the prediction variance Σ θ (z′ t ,t) is a normal distribution.

7. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 1, characterized in that: The specific method of step 6 is: Step 6.1: Encode the physical condition information through the pre-trained conditional encoder to obtain conditional embedding; Step 6.2: Use the cross-attention mechanism to combine the conditional embedding with the new latent variables obtained in step 5 to guide the topology generation; Step 6.3: Use the FiLM layer to introduce the conditional information into the normalization layer of the denoising model to enhance the role of the conditional information.

8. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 2, characterized in that: The specific method of step 8 is: Step 8.1: Use the approximate topology generated by the diffusion process as input to perform a few-step SIMP direct optimization; Step 8.2: Set fixed boundary conditions and mechanical loads for the design domain as initial conditions for the few-step SIMP direct optimization; Step 8.3: Use the finite element analysis method to perform mechanical analysis on the currently input topological structure and calculate the displacement and stress distribution of each design unit; Step 8.4: According to the design update rule in the SIMP method, set the objective function of the few-step SIMP direct optimization, and adjust the material distribution by updating the design variables to maximize the stiffness of the topological structure under the volume or mass constraint, thereby minimizing the flexibility of the topological structure; Step 8.5: Give a volume constraint to the optimization process so that the material volume of the topological structure does not exceed the volume of the design domain; Step 8.6: Repeat the finite element analysis and design variable update until the preset number of iterations is reached, and output the final optimized topology structure.

9. The multi-stage structural topology optimization method based on a potential diffusion model according to claim 8, characterized in that: The objective function set in step 8.4 is shown in the following formula: in, The approximate topology generated in step 7 is is the flexibility value of the approximate topological structure, k e is the stiffness matrix of the design unit; The update of design variables is shown in the following formula: in, is the unit density at the kth iteration, η is the learning rate, is the gradient of the objective function with respect to the density.

Images