A Deep Learning-Based Isogeometric Heat Conduction Simulation Method

Abstract

The present invention discloses an isogeometric heat conduction simulation method based on deep learning, which includes the following steps: Step (1) production of a data set; Step (2) proposing a custom loss function and using it for training; Step (3) training a UNet3+ network enhanced by introducing an ISSA module; Step (4) prediction of the optimal network model and result analysis. By adopting the above technical solution, heat simulation analysis and prediction can be carried out on a complex model with consistent topology, and a smooth and continuous simulation solution can be obtained quickly. On the premise of ensuring accuracy, the simulation time is greatly reduced, which is applicable to the scenario of real-time heat simulation analysis where the shape of the CAD model changes continuously. Compared with PINNs, which have been studied more in recent years, the network model in the present invention only takes 0.013 s to carry out heat simulation analysis and prediction on a new CAD model with the same topology; while the network model in PINNs needs to be trained for about 44 minutes to achieve the same accuracy as the present invention for a new CAD model.

Description

Technical Field

[0001] The present invention relates to the technical field of thermal simulation analysis, and in particular to an isogeometric heat conduction simulation method based on deep learning. Background Art

[0002] Partial differential equations (PDEs) describe numerous physical phenomena and natural laws, playing a crucial role in science, engineering, and economics. Heat conduction is a particularly important PDE. Many industrial processes rely primarily on heat conduction, such as analyzing the internal temperature distribution of metal materials during casting, welding, and forging, as well as design calculations for high-temperature and high-pressure equipment. However, analytical solutions to PDEs are generally unavailable, requiring the use of numerical methods such as finite element analysis and isogeometric analysis for simulation analysis. However, these methods often suffer from high computational complexity and slow computational speeds.

[0003] In recent years, deep learning has developed rapidly and demonstrated outstanding performance in diverse fields, including image, speech, and natural language processing. Given the power of deep neural networks (DNNs) and the successful application of deep learning, deep learning has gradually been applied to other fields of science and engineering, such as computational mechanics. By leveraging DNNs, simulation analysis time has been significantly reduced.

[0004] However, most research on DNN-based simulation analysis is limited to regular rectangular domains. Some work can perform simulation analysis on irregular computational domains, but these domains are not very complex. These studies not only impose significant restrictions on the shape of the simulation domain but also cannot be used in scenarios requiring high real-time performance, as they require retraining the network model for each new CAD model. Therefore, there is an urgent need to develop a method that can rapidly perform thermal simulation analysis on complex CAD models. Summary of the Invention

[0005] The purpose of the present invention is to address the shortcomings of the existing technology and provide an isogeometric heat conduction simulation method based on deep learning. By proposing a loss function that combines the coefficient error and the error of the simulation solution, an enhanced UNet3+ network is trained on a self-made dataset. The obtained optimal model can perform the same thermal simulation analysis and prediction on other complex CAD models with consistent topology, and quickly obtain a continuous and smooth prediction solution.

[0006] In order to solve the above technical problems, the technical solution of the present invention is:

[0007] A deep learning-based isogeometric heat conduction simulation method includes the following steps:

[0008] Step (1) Dataset creation

[0009] 1-1 Produce multiple topologically consistent CAD models,

[0010] 1-2 Perform a thermal simulation analysis with a constant heat source on the CAD model through an isogeometric analysis library to obtain the isogeometric analysis simulation solution,

[0011] 1-3 Preprocess the CAD model and the isogeometric analysis simulation solution of the heat conduction equation to produce a training dataset;

[0012] Step (2) Propose a custom loss function and use it for training

[0013] 2-1 According to the characteristics of the dataset, propose the coefficient error as the loss function,

[0014] 2-2 According to the characteristics of the task, propose the error of the simulation solution as the loss function,

[0015] 2-3 Combine the coefficient error and the error of the simulation solution as the loss function for training;

[0016] Step (3) Train the UNet3+ network enhanced by introducing the ISSA module;

[0017] Step (4) Optimal network model prediction and result analysis.

[0018] Preferably, the preprocessing method in step 1-3 is

[0019] First, normalize the CAD model, limit the CAD model within the range of [0, 1]×[0, 1], find the minimum and maximum values of the x-coordinate and y-coordinate of the CAD model respectively, and calculate the scaling ratio. The expression is as follows:

[0020] s = 1 / max(x max -x min ,y max -y min )

[0021] Thus, the normalized coordinates of the CAD model are obtained. The expression is as follows:

[0022] x n = (x - x min ) × s, y n = (y - y min ) × s

[0023] After obtaining the normalized CAD model, convert it into the input-output format required by the neural network,

[0024] The neural network is a CNN model. The control vertices P of the CAD model ij and the coefficients u in the isogeometric analysis numerical solutionij Convert to matrix format,

[0025] Extract the control vertices from the normalized CAD model, and then according to Simple mapping, through the control vertex P ij of (x ij , y ij ) coordinates to obtain the subscript (r ij in the M×N matrix ij , c ij ),

[0026] After obtaining the subscripts of all control vertices in the matrix, fill different values at the positions of the control vertices in the matrix to obtain matrices with different meanings.

[0027] Preferably, in the step 2-1, the characteristic of the data set is that there are a large number of filled 0s in the output coefficient matrix. The coefficient error is proposed as the loss function, and the expression is as follows:

[0028]

[0029] Among them, is the coefficient predicted by the network, u i is the coefficient in the numerical solution of the isogeometric analysis library, M is the number of coefficients, L u Only calculate the error between the coefficients, not the filled 0s.

[0030] Preferably, in the step 2-2, the task characteristic is to obtain the simulation solution of the heat conduction equation. During the process of training the neural network, the loss function is designed by calculating the error of the simulation solution, and the expression is as follows:

[0031]

[0032] Among them, S represents the number of sampled points on the geometric model. Among them, is the solution predicted by the network, Φ h (x k , y k ) is the isogeometric analysis simulation solution obtained by the isogeometric analysis library. The predicted coefficient is an intermediate link. After obtaining the coefficient, it needs to be linearly combined with the spline basis function to obtain a smooth and continuous simulation solution

[0033] Preferably, in the step 2-3, the expression for training is as follows:

[0034]

[0035] Among them, L u and These two loss functions focus on different aspects. L u is dedicated to reducing the error of the coefficient, while focusing on reducing the error of the simulation solution. α and β represent weights.

[0036] Preferably, in step (3), a self-attention layer is added before the output layer of the UNet3+ network architecture.

[0037] Preferably, in step (4), the testing and analysis are carried out from four aspects, including the coefficient error, the error of the simulation solution, the relative error of the simulation solution, and the a posteriori error of the heat conduction equation.

[0038] Preferably, the calculation methods of the coefficient error and the error of the simulation solution are the same as the formulas in the corresponding loss functions.

[0039] Preferably, the relative error of the simulation solution is the relative error between the network prediction solution and the isogeometric analysis simulation solution obtained from the isogeometric analysis library, and the calculation formula is as follows:

[0040]

[0041] Preferably, the a posteriori error of the heat conduction equation is derived from the heat conduction equation with a constant heat source, and the expression is as follows,

[0042]

[0043] where f(x, y) is the specified right-hand side function in the heat conduction equation.

[0044] The present invention has the following characteristics and beneficial effects:

[0045] By adopting the above technical solution, thermal simulation analysis and prediction can be carried out on complex models with consistent topologies, and smooth and continuous simulation solutions can be obtained quickly. On the premise of ensuring accuracy, the simulation time is greatly reduced, which is applicable to the scenario of real-time thermal simulation analysis where the shape of the CAD model changes continuously. Compared with the PINNs that have been studied more in recent years, the network model in the present invention only takes 0.013 s to carry out thermal simulation analysis and prediction on a new CAD model with the same topology; while the network model in PINNs needs to be trained for about 44 minutes to achieve the same accuracy as the present invention for a new CAD model. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0047] Figure 1 This is the working principle diagram of the embodiment of the present invention;

[0048] Figure 2 It is a display of a spline model with consistent topology, Figure 2 (a) is a human body model, Figure 2 (b) is a flower model;

[0049] Figure 3 To make the control vertices of the normalized spline model into a matrix format;

[0050] Figure 4 It is a schematic diagram of the UNet3+ network architecture enhanced by introducing the ISSA module. Specific implementation manners

[0051] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0052] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by the terms "center", "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation to the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, the features defined with "first", "second", etc. may explicitly or implicitly include one or more of such features. In the description of the present invention, unless otherwise specified, the meaning of "plurality" is two or more.

[0053] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "mounted", "connected", "coupled" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection, an electrical connection; it can be directly connected, or indirectly connected through an intermediate medium, and it can be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific situations.

[0054] The present invention provides a deep learning-based isogeometric heat conduction simulation method, as Figure 1 shown, including the following steps:

[0055] Step (1) Making the dataset: Multiple CAD models with consistent topologies are made. Thermal simulation analysis is carried out on these CAD models by the isogeometric method to obtain simulation solutions, and then data for training is obtained through preprocessing.

[0056] Step (2) Proposing and using a custom loss function for training: According to the characteristics of the dataset, coefficient error is proposed as the loss function; according to the task characteristics, the error of the simulation solution is proposed as the loss function. In the present invention, a combined loss function of the two is used for training.

[0057] Step (3) Training the UNet3+ network enhanced by the ISSA module: The UNet3+ network includes a feature extraction part, an upsampling part, and full-scale skip connections. In the present invention, an ISSA module is added before the output layer of the UNet3+ to obtain global features and enhance the network prediction ability.

[0058] Step (4) Optimal network model prediction and result analysis: The same thermal simulation analysis prediction is carried out on other CAD models with consistent topologies, and the prediction solutions are analyzed through four test metrics.

[0059] In this embodiment, the CAD model takes the B-spline model as an example.

[0060] A further setting of the present invention is that the step (1) includes

[0061] 1-1 Making multiple B-spline models with consistent topologies. Specifically, as Figure 2 shown, in this embodiment, two groups of B-spline models with different topologies are made through the AXL software and a batch program. Figure 2 (a) is a human body model composed of 15 splines, and there are 6300 human body models with different shapes in this group of models. Figure 2 (b) is a flower model composed of 5 splines, and there are 4000 flower models with different shapes in this group of models.

[0062] 1-2 Conducting thermal simulation analysis with a constant heat source on the CAD model through the IGA (isogeometric analysis) library to obtain isogeometric analysis simulation solutions. In the present invention, after normalizing the CAD model, the Gismo library is used to solve the same steady-state heat conduction equation on these models to obtain the IGA simulation solutions of each model.

[0063] It should be noted that the Gismo library is the specific name of the IGA library.

[0064] 1-3 Preprocessing the B-spline model and the isogeometric analysis simulation solutions of the heat conduction equation to make a dataset for training.

[0065] First, we need to normalize the B-spline model and restrict the model within the range of [0, 1] × [0, 1]. Find the minimum and maximum values of the x-coordinate and y-coordinate of the model respectively, and calculate the scaling ratio.

[0066] s = 1 / max(x max - x min , y max - y min ),

[0067] Thus, the coordinates of the normalized model are obtained.

[0068] x n = (x - x min ) × s, y n = (y - y min ) × s.

[0069] After obtaining the normalized CAD model, further processing is carried out to convert them into the input and output formats required by the neural network. The network architecture of the present invention is a CNN model, so the control vertices P ij of the spline model and the coefficients u ij in the IGA numerical solution need to be converted into matrix format.

[0070] As Figure 3 shown, first, the control vertices are extracted from the normalized B-spline model. As shown in the middle figure, the black dots represent the control vertices; then, according to a simple mapping, the subscripts (r ij , c ij , y ij ) of P ij in the M × N matrix are obtained through the (x ij , c ij ) coordinates of the control vertex P

[0071] As shown in the rightmost figure, the pure black 'x' indicates that this position is a control vertex, and the remaining light gray dots represent filled 0s.

[0072] After obtaining the subscripts of all control vertices in the matrix, different values are filled in the positions of the control vertices in the matrix to obtain matrices with different meanings. Fill 1 at the control vertices to obtain a 0-1 matrix representing the positions of the control vertices; fill in the x-coordinate of the control vertex at the control vertex to obtain the x-coordinate matrix of the control vertex; the y-coordinate matrix of the control vertex is obtained in the same way. The control vertices and the coefficients are in one-to-one correspondence. Fill in the coefficient values at the control vertices to obtain the coefficient matrix.

[0073] In step (2), a custom loss function is proposed and trained. According to the characteristics of the data set, the coefficient error is proposed as the loss function; according to the characteristics of the task, the error of the simulation solution is proposed as the loss function. The present invention uses a combined loss function of the two for training.

[0073] A further setting of the present invention, the step (2) includes

[0074] 2-1 According to the characteristics of the data set, coefficient error is proposed as the loss function

[0075] In deep learning, the most commonly used loss function for regression tasks is the mean squared error (MSE), which is obtained by calculating the mean of the squared errors between one-to-one corresponding data points. The formula is as follows

[0076]

[0077] where is the output of the network, y i is the true value, N represents the number of data. In the present invention, the size of N is 128 2 .

[0078] However, considering the particularity of the data set in this embodiment, that is, there are a large number of filled 0s in the output coefficient matrix. In fact, what we need is only the coefficients among them. And the calculation of MSE includes these filled 0s, so a lot of effort is required to fit these filled 0s when training the neural network. Therefore, we modify the loss function as follows

[0079]

[0080] where is the coefficient predicted by the network, u i is the coefficient in the numerical solution of the Gismo library, and M is the number of coefficients. L u only calculates the error between coefficients and does not calculate the filled 0s, enabling the neural network to concentrate on fitting the values of the coefficients to improve the prediction accuracy of the network. Compared with the network model trained with L MSE as the loss function, the network model trained with L u reduces the relative error of the numerical solution from 13% to 7.67%, with a reduction of 41%.

[0081] 2-2 According to the task characteristics, the error of the simulation solution is proposed as the loss function

[0082] Returning to the task itself, the ultimate goal of the present invention is to obtain the simulation solution of the heat conduction equation. Predicting the coefficient is an intermediate link. After obtaining the coefficient, it needs to be linearly combined with the spline basis function to obtain a continuous and smooth simulation solution Therefore, during the process of training the neural network, the loss function can be designed by calculating the error of the simulation solution

[0083]

[0084] where S represents the number of sampling points on the geometric model is the solution predicted by the network, Φh (x k , y k ) is the IGA simulation solution obtained by Gismo. Compared with the network model trained with L MSE as the loss function, the network model trained with reduced the relative error of the simulation solution from 13% to 7.31%, a decrease of 44%.

[0085] Training with the loss function that combines the coefficient error and the error of the simulation solution

[0086] L u and These two loss functions focus on different perspectives. L u focuses on reducing the error of the coefficient, focuses on reducing the error of the simulation solution. Therefore, the present invention adopts the method of combining these two loss functions for training

[0087]

[0088] u and When training the network, not only does it not affect the training results of each other, but it also combines the advantages of these two loss functions, achieving the effect of 1 + 1 > 2.

[0089] For a further setting of the present invention, in step (3), the ISSA module - enhanced UNet3+ network is introduced during training to enhance the network prediction accuracy.

[0090] Specifically, as Figure 4 shown, it includes an input layer and an output layer. The network model used in the present invention is composed of 12 layers and corresponding convolutional blocks. Each convolutional block has a similar structure: convolutional calculation, batch normalization, and a non - linear activation function. Among them, the Down - sampling module includes max - pooling and a convolutional block, and the Up - sampling module includes an up - sampling operation and a convolutional block.

[0091] The biggest feature of UNet3+ is the full - scale skip connection. As Figure 4 shown by the dotted line, it combines low - level features and high - level features, making full use of the feature information extracted in each stage. The skip connection module is composed of a max - pooling or up - sampling operation and a convolutional block.

[0092] In addition to the classic UNet3+ architecture, we added a self - attention (SA) layer before the output layer. See Figure 4Convolution is a local operation. Therefore, in the CNN architecture, it is necessary to introduce SA (Self-Attention) for obtaining long-range dependencies of pixel points to obtain more global features. The present invention introduces the ISSA module, that is, the accelerated SA module, which can effectively obtain global features and improve the network prediction accuracy. In the experiment, the network using the ISSA module reduced various errors by about 10%.

[0093] In this embodiment, the input size of the network model is 128 2 ×3, and the output size is 128 2 ×1. The 0-1 matrix, x-coordinate matrix, and y-coordinate matrix of the B-spline model control vertices in the dataset preprocessing are respectively input into the network as one channel, and then through the encoding and decoding processes and skip connection operations, a coefficient matrix of one channel is finally output. Coefficient values are extracted from the coefficient matrix to construct a smooth and continuous predicted solution for the heat conduction equation

[0094] A further setting of the present invention, step (4) optimal network model prediction and result analysis, perform the same thermal simulation analysis and prediction on other CAD models with the same topology, and analyze the predicted solution through four test indicators

[0095] Specifically, train the network on the training set to obtain the optimal model, and then the same thermal simulation analysis and prediction can be performed on other CAD models with the same topology

[0096] For the prediction results of the network model, the present invention tests from four aspects: coefficient error, error of the simulation solution, relative error of the simulation solution, and posterior error of the heat conduction equation. The calculation methods of the coefficient error and the error of the simulation solution are the same as the formulas in step (2), and they can reflect the role of the loss function. The relative error of the numerical solution is the relative error between the network predicted solution and the IGA simulation solution, which is convenient for comparing the prediction effects of different network models. The calculation formula is as follows

[0097]

[0098] This article solves the steady-state heat conduction equation, and it is easy to derive the explicit representation of its posterior error

[0099]

[0100] Among them, f(x, y) is the specified right-hand side function in the heat conduction equation. The posterior error of the PDEs can indicate whether the obtained simulation solution can satisfy the physical laws of the PDEs themselves

[0101] In this embodiment, two geometric models with different topologies are made, and the heat conduction equations of two different heat sources are solved. We conduct experiments on solving the two equations on two data sets respectively, proving that the present invention can accurately conduct thermal simulation analysis and prediction on other CAD models with consistent topologies. Among them, when conducting thermal simulation analysis of a constant heat source on a flower model composed of 5 splines, the relative error between the predicted solution and the IGA simulation solution is only 2.36%, reaching an extremely low level, with high prediction accuracy. Moreover, it only takes 0.013 s to conduct thermal simulation analysis and prediction on a new CAD model with a consistent topology, greatly shortening the time of thermal simulation analysis and prediction and improving the prediction efficiency.

[0102] The embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, without departing from the principle and spirit of the present invention, various changes, modifications, substitutions, and variations to these embodiments including components still fall within the protection scope of the present invention.

Claims

1. A deep learning-based isogeometric heat conduction simulation method, characterized in that, It includes the following steps: Step (1) Production of the dataset 1-1 Produce multiple CAD models with consistent topologies, 1-2 Perform thermal simulation analysis with a constant heat source on the CAD models through an isogeometric analysis library to obtain isogeometric analysis simulation solutions, 1-3 Preprocess the CAD models and the isogeometric analysis simulation solutions of the heat conduction equation to produce a training dataset; Step (2) Propose a custom loss function and use it for training 2-1 According to the characteristics of the dataset, propose the coefficient error as the loss function, The characteristics of the dataset are that there are a large number of filled 0s in the output coefficient matrix. Propose the coefficient error as the loss function, and the expression is as follows: Among them, is the coefficient predicted by the network, and u i is the coefficient in the numerical solution of the isogeometric analysis library. M is the number of coefficients, and L u only calculates the error between coefficients and does not calculate the filled 0; 2-2 According to the task characteristics, propose the error of the simulation solution as the loss function, The task characteristics are to obtain the simulation solution of the heat conduction equation. During the process of training the neural network, design the loss function by calculating the error of the simulation solution, and the expression is as follows: Among them, S represents the number of points sampled on the geometric model, where is the solution predicted by the network, and Φ h (x k , y k ) is the isogeometric analysis simulation solution obtained from the isogeometric analysis library. The prediction coefficient is an intermediate link. After obtaining the coefficient, it needs to be linearly combined with the spline basis function to obtain a smooth and continuous simulation solution 2-3 Combine the coefficient error and the error of the simulation solution as the loss function for training; The expression for training is as follows: Among them, L u and These two loss functions focus on different perspectives. L u is dedicated to reducing the error of the coefficient, while focuses on reducing the error of the simulation solution. α and β represent weights; Step (3) Train the UNet3+ network enhanced by introducing the ISSA module, and add a self-attention layer before the output layer of the UNet3+ network architecture; Step (4) Optimal network model prediction and result analysis, which are tested and analyzed from four aspects, including coefficient error, error of the simulation solution, relative error of the simulation solution, and posterior error of the heat conduction equation.

2. The isogeometric heat conduction simulation method based on deep learning according to claim 1, wherein The preprocessing method in step 1-3 is First, normalize the CAD models, limit the CAD models within the range of [0, 1]×[0, 1], find the minimum and maximum values of the x and y coordinates of the CAD models respectively, and calculate the scaling ratio. The expression is as follows: s = 1 / max(x max - x min , y max - y min ), where s represents the scaling ratio, max(·) represents the maximum function, and x min and x max represent the minimum and maximum values of the x - coordinate of the CAD model respectively, and y min and y max represent the minimum and maximum values of the y - coordinate of the CAD model respectively; Thus, the normalized coordinates (x n , y n ) of the CAD model are obtained, and the expression is as follows: x n =(x - x min )×s,y n =(y - y min )×s After obtaining the normalized CAD models, convert them into the input-output format required by the neural network, The neural network is a CNN model, which converts the control vertices P of the CAD model ij and the coefficients u in the numerical solution of isogeometric analysis ij into a matrix format. Extract the control vertices from the normalized CAD model, and then according to a simple mapping, obtain the subscript (r ij , c ij ) of P ij in the M×N matrix through the (x ij , y ij ) coordinates of the control vertex P ij . After obtaining the subscripts of all control vertices in the matrix, fill in different values at the positions of the control vertices in the matrix to obtain matrices with different meanings.

3. The isogeometric heat conduction simulation method based on deep learning according to claim 1, characterized in that The calculation methods of the coefficient error and the error of the simulation solution are the same as the formulas in the corresponding loss functions.

4. The isogeometric heat conduction simulation method based on deep learning according to claim 1, characterized in that The relative error of the simulation solution is the relative error between the network prediction solution and the isogeometric analysis simulation solution obtained by the isogeometric analysis library. The calculation formula is as follows:

5. The isogeometric heat conduction simulation method based on deep learning according to claim 1, characterized in that The posterior error of the heat conduction equation is derived from the heat conduction equation with a constant heat source, and the expression is as follows, where f(x, y) is the specified right-hand side function in the heat conduction equation.

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