Bubble dynamic boundary reconstruction method in machine learning driven gas-liquid two-phase flow simulation

Abstract

This invention discloses a machine learning-driven method for reconstructing the dynamic boundary of bubbles in gas-liquid two-phase flow simulation. The method includes: recording bubble videos and binarizing them; meshing and preprocessing to construct feature and label arrays; training a spatiotemporal convolutional graph neural network to obtain a prediction model; performing numerical calculations, dividing the solution region into meshes, determining the mesh containing the bubble boundary, deriving volume fraction values ​​and inputting them into the model, outputting the predicted interface position, calculating the flux based on the velocity field, and updating the next volume fraction value; repeating this process until all predicted interface positions are predicted, thus achieving dynamic bubble boundary reconstruction. This invention effectively addresses the problems of limited accuracy and high computational cost in traditional numerical methods for solving two-phase flow problems. By coupling machine learning with traditional numerical methods, it can accurately predict the dynamic boundary position of bubbles at each time step, providing support for numerical research on two-phase flow problems.

Description

Technical Field

[0001] This invention relates to a bubble dynamic boundary reconstruction method, which relates to the field of multiphase flow numerical simulation, and specifically to a machine learning-driven bubble dynamic boundary reconstruction method in gas-liquid two-phase flow simulation. Background Technology

[0002] Gas-liquid two-phase flow refers to a flow system involving the interaction of gas and liquid phases. It is widely found in nuclear power, energy, machinery, aerospace, and everyday life, such as in chemical reactors, bubbling beds, heat exchangers, and high-speed pumps. For example, in the thermal design of thermodynamic systems, boiling heat transfer within the heat exchange channels is typically required to fully utilize latent heat and significantly enhance heat transfer capacity. Similarly, in high-speed pump-driven flow systems, the pump inlet often involves bubbly inflow conditions, affecting flow field evolution and energy conversion characteristics. These are all typical industrial scenarios of gas-liquid two-phase flow. Currently, numerical methods are commonly used to study two-phase flow problems, and the key and challenging aspect lies in accurately capturing the interface between the two phases. Traditional two-phase interface tracking methods include the volume of fluid (VOF), level set (LS), and coupled volume-of-fluid and level set (VOSET) methods. However, these methods not only suffer from numerical dissipation and limited accuracy, but also have problems such as complex solution processes and high computational resource consumption when reconstructing the interface.

[0003] When performing interface reconstruction, multiple equations need to be solved and cases need to be classified, which consumes a lot of computational resources. At the same time, the interface reconstruction method uses a straight line to approximate the interface curve, which will introduce errors into the calculated mass flow rate. These errors are more obvious in fast and dense bubble flow situations (such as nuclear power plant cooling pipes). Summary of the Invention

[0004] To address the problems existing in the background technology, this invention provides a machine learning-driven method for reconstructing the dynamic boundary of bubbles in gas-liquid two-phase flow simulation. The continuous development of Artificial Intelligence (AI) and machine learning technologies has provided new solutions to problems encountered by traditional numerical methods. Machine learning can effectively extract the mapping relationships between multiple parameters and objectives from data, solving nonlinear and high-dimensional problems, and is applicable to fields such as turbulence modeling, aerodynamics, and image processing. In two-phase flow, machine learning methods can be used to improve phase interface tracking, reduce computational resource consumption, and improve simulation accuracy, especially for challenging scenarios such as dense bubble flow fields. This provides a more promising approach for solving two-phase flow problems in the future.

[0005] The technical solution adopted in this invention is:

[0006] The machine learning-driven bubble dynamic boundary reconstruction method in gas-liquid two-phase flow simulation of the present invention includes:

[0007] S1: Use a high-speed camera to record video of bubbles in a liquid, and then binarize each video frame to obtain several consecutive binarized bubble images.

[0008] S2: Divide each binarized bubble image into a grid, construct an image information array Z based on the image information of each grid, and construct a feature array S and a label array L after preprocessing the image information array Z.

[0009] S3: Establish a spatiotemporal convolutional graph neural network. Use the feature array S and label array L as input and output respectively to train the spatiotemporal convolutional graph neural network using supervised learning. After training, obtain a data-driven bubble dynamic boundary prediction model.

[0010] S4: Construct a physical model for the two-phase flow scenario, divide the computational region in the physical model into grids, determine the grid where the bubble boundary is located based on the volume fraction value of each grid at the current time step, perform numerical simulation on each grid where the bubble boundary is located to obtain the volume fraction value and velocity field information at the current time step, and export the volume fraction values ​​of each grid where the bubble boundary is located and its surrounding grids before the current time step.

[0011] S5: Input the volume fraction values ​​of each bubble boundary grid and its surrounding grids before the current time step in step S4 into the data-driven bubble dynamic boundary prediction model for processing. After processing, the data-driven bubble dynamic boundary prediction model outputs the predicted interface position of each bubble boundary grid at the current time step. Combine the predicted interface position with the velocity field information to calculate the flux. Update the volume fraction value of the bubble boundary grid at the next time step according to the flux.

[0012] S6: Repeat steps S4-S5 until the grid containing the bubble boundary in all time steps to be solved has predicted the interface position, thus realizing the dynamic boundary reconstruction of the bubble.

[0013] In step S1, a high-speed camera is used to record video of bubbles in the liquid. All video frames from time t to t+T are extracted from the video. Each video frame is binarized so that the pixel value of the liquid phase in the video frame is 0 and the pixel value of the gas phase is 255, thereby obtaining several consecutive binarized bubble images from time t to t+T.

[0014] In step S2, each binarized bubble image is divided into a grid, uniformly dividing the binarized bubble image into... A rectangular grid, ,in, and These represent the vertical and horizontal pixel counts of the image, respectively. and Set the grid width and height respectively; then create a shape with ( An empty array, The total time of the video; the image information array Z is constructed by inputting the image information of each grid of each binarized bubble image into an empty array.

[0015] The image information includes the volume fraction values ​​of the grid and the interface position. The interface position includes the intersection position information of the four sides of the grid, as detailed below:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] in, The grid in the i-th row and j-th column of the binarized bubble image The volume fraction value, , , and The grid cells in the i-th row and j-th column of the binarized bubble image are respectively... Information on the interface positions at the top, bottom, left, and right sides.

[0022] Input the five sets of information from the grid into the corresponding positions in the empty array in sequence, and repeat this step continuously until the entire image has been traversed.

[0023] In step S2, the image information array Z is preprocessed to construct the feature array S and the label array L, as detailed below:

[0024] First, create a shape of ( The feature empty array is defined as follows: ...

[0025] Then create a shape of ( The label array is empty. All grids and interface position data at all times in the image information array Z are obtained and input into the label array to construct the final label array L.

[0026] In step S3, the spatiotemporal convolutional graph neural network includes several spatiotemporal blocks connected in sequence and a fully connected layer. When the input of the spatiotemporal convolutional graph neural network is processed by each spatiotemporal block, the input and output of each spatiotemporal block are used as the input of the next spatiotemporal block.

[0027] Construct a spatiotemporal convolutional graph neural network with an undirected and unweighted structure. Define a network, where the finite set V represents the number of grid cells. E represents the edge set and the adjacency matrix. This indicates the degree of correlation between grids.

[0028] The spatiotemporal block includes a spatial attention mechanism, a convolutional layer (Conv), and a graph convolutional network (GCN) connected in sequence.

[0029] In step S4, the volume fraction value of each grid at the current time step is used as the basis for the division. ,when If the grid at the current time step is determined to be the grid containing the bubble boundary, then for the current time step n, the volume of fluid (VOF) method is used to solve for the grid containing the bubble boundary, obtaining the first T of the grid containing the bubble boundary and the surrounding a×a range of grids. d The volume fraction values ​​at each time step are used as input data to drive the bubble dynamic boundary prediction model, which then outputs the predicted interface position after processing.

[0030] The beneficial effects of this invention are:

[0031] 1) By utilizing artificial intelligence to learn the distribution patterns of the gas-liquid interface from actual bubble motion images, and through iterative solving coupled with numerical methods, the position of the dynamic bubble interface can be accurately predicted. Compared with traditional interface reconstruction methods, it avoids the approximation error of "substituting straight lines for curves" and considers spatiotemporal dependence, thus improving accuracy.

[0032] 2) It avoids the operations of solving high-order equations and classifying cases in the process of reconstructing the interface using traditional numerical methods, which can significantly reduce the computational difficulty, reduce computational resources, and reduce R&D and testing costs while ensuring accuracy.

[0033] In summary, this invention addresses the problems of limited accuracy in tracking the two-phase interface and high computational cost encountered in traditional numerical methods for solving two-phase flow problems. By coupling machine learning with traditional numerical methods, it can accurately predict the dynamic boundary position of bubbles at each time step, providing support for numerical research on two-phase flow problems. Attached Figure Description

[0034] Figure 1 This is a schematic diagram of the fluid volume method (VOF).

[0035] Figure 2 This is a schematic diagram of the PLIC method, where, Figure 2 (a) is a schematic diagram of the original interface. Figure 2 (b) is a schematic diagram of the reconstructed interface;

[0036] Figure 3 This is a schematic diagram of a spatiotemporal convolutional graph neural network framework;

[0037] Figure 4 This is a binarized image from an embodiment of the present invention;

[0038] Figure 5 This is a flowchart of the coupled solution process of the present invention. Detailed Implementation

[0039] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0040] like Figure 5 As shown, the machine learning-driven bubble dynamic boundary reconstruction method in gas-liquid two-phase flow simulation of the present invention includes:

[0041] S1: Record video of bubbles in a liquid using a high-speed camera. Binarize each video frame to obtain several consecutive binarized bubble images, such as... Figure 4 As shown.

[0042] In step S1, a high-speed camera is used to record video of bubbles in the liquid. All video frames from time t to t+T are extracted from the video. Each video frame is binarized so that the pixel value of the liquid phase in the video frame is 0 and the pixel value of the gas phase is 255, thereby obtaining several consecutive binarized bubble images from time t to t+T.

[0043] S2: Divide each binarized bubble image into a grid, construct an image information array Z based on the image information of each grid, and construct a feature array S and a label array L after preprocessing the image information array Z.

[0044] In step S2, each binarized bubble image is divided into a grid, uniformly dividing the binarized bubble image into... A rectangular grid, ,in, and These represent the vertical and horizontal pixel counts of the image, respectively. and Set the grid width and height respectively; then create a shape with ( An empty array, The total time of the video; the image information array Z is constructed by inputting the image information of each grid of each binarized bubble image into an empty array.

[0045] The image information includes the volume fraction values ​​of the grid and the interface position. The interface position includes the location information of the intersection of the grid's four sides, as detailed below:

[0046]

[0047]

[0048]

[0049]

[0050]

[0051] in, The grid in the i-th row and j-th column of the binarized bubble image The volume fraction value, , , and The grid cells in the i-th row and j-th column of the binarized bubble image are respectively... Information on the interface positions at the top, bottom, left, and right sides.

[0052] Input the five sets of information from the grid into the corresponding positions in the empty array in sequence, and repeat this step continuously until the entire image has been traversed.

[0053] In step S2, the image information array Z is preprocessed to construct the feature array S and the label array L, as follows:

[0054] First, create a shape of ( The feature empty array is defined as follows: ...

[0055] Then create a shape of ( The label array is empty. All grids and interface position data at all times in the image information array Z are obtained and input into the label array to construct the final label array L.

[0056] S3: Establish a spatiotemporal convolutional graph neural network. Use the feature array S and label array L as input and output respectively to train the spatiotemporal convolutional graph neural network using supervised learning. After training, obtain a data-driven bubble dynamic boundary prediction model.

[0057] In step S3, the spatiotemporal convolutional graph neural network includes several spatiotemporal blocks connected sequentially and a fully connected layer. When the input of the spatiotemporal convolutional graph neural network is processed through each spatiotemporal block, both the input and output of each spatiotemporal block serve as the input of the next spatiotemporal block, such as... Figure 3 As shown.

[0058] Construct a spatiotemporal convolutional graph neural network with an undirected and unweighted structure. Define a network, where the finite set V represents the number of grid cells. E represents the edge set and the adjacency matrix. This indicates the degree of correlation between grids.

[0059] The spatiotemporal block consists of a spatial attention mechanism, a convolutional layer (Conv), and a graph convolutional network (GCN) connected in sequence.

[0060] Since data can be correlated at different times or under different circumstances, an attention mechanism is used to assign importance to the data. The adjusted, more valuable information is then input into the spatiotemporal convolution module to capture the spatial dependence of the neighborhood and the temporal dependence of the immediate time.

[0061] Spatial convolutional layer: Based on the strong spatial correlation between grid cells, the features of each grid cell can be regarded as signals on the graph. Therefore, at each time step, the model chooses graph convolution operations to capture spatial dependencies. In this way, the information of each grid cell depends on the information of its neighboring grid cells for updating. Temporal convolutional layer: After capturing the spatial dependencies of each grid cell on the graph using graph convolution operations, further stacking convolutions in the temporal dimension is needed to update the output by merging information from adjacent time periods. The spatiotemporal attention mechanism and spatiotemporal convolutional layers form spatiotemporal blocks. Stacking multiple spatiotemporal blocks can better capture the spatiotemporal features of the data. A fully connected layer is attached to ensure that the input and the predicted output have the same dimension and shape. The activation function of the fully connected layer is also ReLU.

[0062] In practice, all spatiotemporal convolutional layers in the model use 64 convolutional kernels, with a batch size of 64 and a learning rate of 0.0001. The mean squared error (MSE) between the predicted and actual values ​​is used as the loss function. The randomly constructed dataset is divided into training and test sets, with the training set comprising 80% of the original dataset. Using the feature matrix S as input and the label matrix L as output, the spatiotemporal convolutional graph neural network model is trained using supervised learning to obtain the data-driven bubble dynamic boundary prediction model.

[0063] S4: For specific two-phase flow scenarios (such as boiling heat transfer), a physical model is constructed. The computational domain in the physical model is divided into grids. The grid where the bubble boundary is located is determined based on the volume fraction value of each grid at the current time step. After numerical simulation of each grid where the bubble boundary is located, the volume fraction value and velocity field information of each grid at the current time step are obtained. At the same time, the volume fraction values ​​of each grid where the bubble boundary is located and its surrounding grids before the current time step are exported.

[0064] In step S4, the volume fraction value of each grid at the current time step is determined based on the partitioned data. ,when If the grid at the current time step is determined to be the grid containing the bubble boundary, then for the current time step n, the volume of fluid (VOF) method is used to solve for the grid containing the bubble boundary, obtaining the first T of the grid containing the bubble boundary and the surrounding a×a range of grids. d The volume fraction values ​​at each time step are used as input data to drive the bubble dynamic boundary prediction model, which then outputs the predicted interface position after processing.

[0065] The method of this invention uses the predicted interface position predicted by the model instead of the simulated interface position obtained after interface reconstruction using the Piecewise Linear Interface Calculation (PLIC) method.

[0066] Solve the VOF function, for the current time step n, obtain the first T of the grid C and the surrounding a×a range of grids. d The volume fraction values ​​at each time step are input into the trained model to predict the current grid. Value, combined with mesh boundary velocity value Calculate the volume fraction flux of the principal phase fluid at each grid boundary, and then update the volume fraction at time step n+1. .

[0067] like Figure 1 As shown, the fluid volume fraction (VOF) method uses fluid volume fraction. This represents the volume fraction of a fluid within a given cell, with values ​​ranging from 0 to 1. This indicates that the current mesh does not contain the fluid; This indicates that the current grid is filled with this fluid; This indicates that the current grid contains a gas-liquid interface. In this case, the fluid volume flux at the grid boundary can be calculated after boundary reconstruction.

[0068] like Figure 2 (a) and Figure 2 As shown in (b), the PLIC method reconstructs the interface by using the idea of ​​"replacing curves with straight lines" to construct straight lines with arbitrary slopes within each cell grid to replace the actual curved surface interface. The steps can be summarized into three steps. Representing the volume fraction of the current computational mesh (i,j), the first step is to calculate the interface normal vector using the volume fractions of the surrounding meshes. The first step involves determining the interface position using the angle β between the interface and the x-axis, and the second step uses the angle and volume fraction to determine the interface location. Fluid morphology can be approximated into 16 types, which are later simplified to 4 types through flipping and symmetry. The third step calculates the fluid mass flow rate through the grid boundary in the current time step based on the grid boundary velocity and interface position, and updates the volume fraction of this grid. .

[0069] S5: Input the volume fraction values ​​of each bubble boundary grid and its surrounding grids before the current time step in step S4 into the data-driven bubble dynamic boundary prediction model for processing. After processing, the data-driven bubble dynamic boundary prediction model outputs the predicted interface position of each bubble boundary grid at the current time step. Combine the predicted interface position with the velocity field information to calculate the flux. Update the volume fraction value of the bubble boundary grid at the next time step according to the flux.

[0070] S6: Repeat steps S4-S5 until the grid containing the bubble boundary in all time steps to be solved has predicted the interface position, thus realizing the dynamic boundary reconstruction of the bubble.

[0071] Specific embodiments of the present invention are as follows:

[0072] Step 10) Construct a bubble experiment platform, including a glass tube, a bubble generator, LED lights, a high-speed camera, etc. Record the bubble behavior with the high-speed camera under LED illumination for 20,000 seconds. Use image processing to convert the captured images into... Figure 4 The binarized image shown has 1800×900 pixels in both the horizontal and vertical directions.

[0073] Step 20) Specify the grid size as 20×20, that is, divide it into quadrilateral grids with a length and width of 20 pixels each, then the shape of Z is (90,45,5,20000).

[0074] Step 301) Starting from grid (0,0), obtain five sets of data: the volume fraction value within the current grid and the positions of the interfaces at the top, bottom, left, and right edges. Input these data into Z.

[0075] Step 302) Traverse the entire image.

[0076] Step 303) Replace the images at different times, repeat the steps continuously, and expand Z.

[0077] Step 401) Set the sliding window size to 3×3, create a feature array S with shape (20000,4050,9) and a label array L with shape (20000,4050,4).

[0078] Step 402) Obtain the volume fraction values ​​of all grids within the current sliding window region at all time points, input them into the first line of S, and obtain the l values ​​of the intermediate grids. t l b l l l r Enter the value into the first line of L.

[0079] Step 403) Slide the window to the next position.

[0080] Step 404) Repeat the steps until the entire array Z has been traversed.

[0081] Step 501) Take the data in the same row of arrays S and L as a group of data, and randomly divide the dataset into training set and test set, with the training set accounting for 80%.

[0082] Step 502) Construct as follows Figure 3 The bubble boundary prediction spatiotemporal convolutional network model shown is loaded with a training set, with a batch size of 64. The data is input into the model in batches for training. The loss function is set to mean squared error (MSE), and the learning rate is set to 0.0001. The first round of training begins, i.e., epoch=1. The specific training steps are as follows:

[0083] 1) Input the training data into the spatiotemporal attention mechanism module to adaptively capture the dynamic correlation between the time and spatial dimensions of the grid.

[0084] 2) More valuable information is input into the spatiotemporal convolutional network layer to obtain the output of the first layer, thus completing the first spatiotemporal block.

[0085] 3) Input the output of the first spacetime block into the next time attention mechanism module, and repeat the above steps until all spacetime blocks have been processed.

[0086] 4) Input the output of the last spatiotemporal block into the fully connected layer to obtain a prediction result with the same shape and dimension as the output data.

[0087] Step 503) Backpropagation of the model updates the parameters.

[0088] Step 504) Input the test set data into the training model for this round of testing, calculate the loss and save the model training parameters.

[0089] Step 505) Repeat steps 502-504 until the training ends when the training termination condition is met or the number of training rounds is greater than the maximum value.

[0090] Step 60) Figure 5 The process shown establishes a mathematical model for the computational domain and performs numerical calculations. For the current time step n, iteratively checks whether the cell grid C(i, j) is the cell containing the bubble boundary. If so, it obtains the historical input features of the previous 20 time steps. The input model predicts the l of the current grid. t l b l l l r Value, combined with the mesh boundary velocity value u t u b u l u r Finally, the flow rate of the fluid on each side is calculated, and the volume fraction at time step n+1 is updated.

[0091] By coupling machine learning with numerical solutions and training the model with real samples to predict the bubble interface position, errors caused by interface reconstruction can be avoided and computational resources can be reduced.

[0092] Finally, it should be noted that the embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the invention. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the invention. All technical solutions obtained through equivalent substitutions or transformations fall within the protection scope of the present invention.

Claims

1. A machine learning-driven method for reconstructing dynamic bubble boundaries in gas-liquid two-phase flow simulation, characterized in that, include: S1: Use a high-speed camera to record video of bubbles in a liquid, and then binarize each video frame to obtain several consecutive binarized bubble images. S2: Divide each binarized bubble image into a grid, construct an image information array Z based on the image information of each grid, and construct a feature array S and a label array L after preprocessing the image information array Z; S3: Establish a spatiotemporal convolutional graph neural network. Use the feature array S and the label array L as input and output respectively to train the spatiotemporal convolutional graph neural network using supervised learning. After training, obtain a data-driven bubble dynamic boundary prediction model. S4: Construct a physical model for the two-phase flow scenario, divide the computational region of the physical model into grids, obtain the grid where the bubble boundary is located based on the volume fraction value of each grid at the current time step after division, perform numerical simulation on each grid where the bubble boundary is located to obtain the volume fraction value and velocity field information at the current time step, and export the volume fraction value of each grid where the bubble boundary is located and its surrounding grids before the current time step. S5: Input the volume fraction values ​​of each bubble boundary grid and its surrounding grids before the current time step in step S4 into the data-driven bubble dynamic boundary prediction model for processing. After processing, the data-driven bubble dynamic boundary prediction model outputs the predicted interface position of each bubble boundary grid at the current time step. Combine the predicted interface position with the velocity field information to calculate the flux. Update the volume fraction value of the bubble boundary grid at the next time step according to the flux. S6: Repeat steps S4-S5 until the grid containing the bubble boundary in all time steps to be solved has predicted the predicted interface position, thus realizing the dynamic boundary reconstruction of the bubble. In step S2, each binarized bubble image is divided into a grid, uniformly dividing the binarized bubble image into... A rectangular grid, ,in, and These represent the vertical and horizontal pixel counts of the image, respectively. and Set the grid width and height respectively; then create a shape with ( An empty array, The total number of moments in the video; the image information array Z is constructed by inputting the image information of each grid of each binarized bubble image into an empty array; The image information includes the volume fraction values ​​of the grid and the interface position. The interface position includes the intersection position information of the four sides of the grid, as detailed below: in, The grid in the i-th row and j-th column of the binarized bubble image The volume fraction value, , , and The grid cells in the i-th row and j-th column of the binarized bubble image are respectively... The location information of the interface at the top, bottom, left, and right sides; In step S2, the image information array Z is preprocessed to construct the feature array S and the label array L, as detailed below: First, create a shape of ( The feature empty array is obtained by using a sliding window of size a×a to traverse the image information array Z, obtaining the volume fraction values ​​of all grids and all time points within the current sliding window area and inputting them into the feature empty array. The sliding window moves to the next position, and the steps are repeated until the entire image information array Z is traversed, thus constructing the final feature array S. Then create a shape of ( The label array is empty. All grids and interface position data at all times in the image information array Z are obtained and input into the label array to construct the final label array L.

2. The method for reconstructing dynamic bubble boundaries in machine learning-driven gas-liquid two-phase flow simulation according to claim 1, characterized in that: In step S1, a high-speed camera is used to record video of bubbles in the liquid. All video frames from time t to t+T are extracted from the video. Each video frame is binarized so that the pixel value of the liquid phase in the video frame is 0 and the pixel value of the gas phase is 255, thereby obtaining several consecutive binarized bubble images from time t to t+T.

3. The method for reconstructing dynamic bubble boundaries in machine learning-driven gas-liquid two-phase flow simulation according to claim 1, characterized in that: In step S3, the spatiotemporal convolutional graph neural network includes several spatiotemporal blocks connected in sequence and a fully connected layer. When the input of the spatiotemporal convolutional graph neural network is processed by each spatiotemporal block, the input and output of each spatiotemporal block are used as the input of the next spatiotemporal block.

4. The method for reconstructing dynamic bubble boundaries in machine learning-driven gas-liquid two-phase flow simulation according to claim 3, characterized in that: The spatiotemporal block comprises a spatial attention mechanism, a convolutional layer (Conv), and a graph convolutional network (GCN) connected in sequence.

5. The method for reconstructing dynamic bubble boundaries in machine learning-driven gas-liquid two-phase flow simulation according to claim 1, characterized in that: In step S4, the volume fraction value of each grid at the current time step is used as the basis for the division. ,when If the grid at the current time step is determined to be the grid containing the bubble boundary, then for the current time step n, the Volume of Fluid (VOF) method is used to solve for the grid containing the bubble boundary, obtaining the first T of the grid containing the bubble boundary and the surrounding a×a range of grids. d The volume fraction values ​​at each time step are used as input data to drive the bubble dynamic boundary prediction model, which then outputs the predicted interface position after processing.

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