Turbine part elastic digital twinning virtual-real interaction method combining geometric features and Fourier neural operators

Abstract

The invention discloses a turbine part elastic digital twinning virtual-real interaction method combining geometric features and Fourier neural operators. The method comprises the following steps: S1, setting material attributes and boundary conditions; s2, finite element data generation and format conversion; s3, constructing and training a prediction model based on a geometric perception Fourier operator; and S4, model prediction and verification. According to the method, finite element software and AI prediction are combined, the turbine parts are subjected to self-defined Fourier transform and geometric coding technologies, the problems that an existing turbine part analysis method is complex in calculation and complex geometric features are effectively processed are solved, a traditional elastic analysis method is optimized, and the method has the advantages of being high in reliability and high in reliability. The method is a technology for improving the analysis precision while reducing the calculation complexity, and is suitable for solving complex geometrical shapes and physical phenomena. The method has a wide application prospect, and can provide efficient calculation support for design optimization, performance evaluation and fault detection.

Description

Technical Field

[0001] The present invention belongs to the field of intelligent simulation and relates to a virtual-reality interaction method of a digital twin of the elasticity of a turbine part, and specifically relates to a virtual-reality interaction method of a digital twin of the elasticity of a turbine part that combines geometric features with a Fourier neural operator. Background Art

[0002] Turbine parts are key components of modern mechanical systems and are widely used in high-tech fields such as aircraft engines, gas turbines, and wind turbines. These turbine parts usually face extreme working environments, such as high temperature, high pressure, and high-speed rotation, and are subjected to complex mechanical and thermal loads. Therefore, accurately predicting the elastic deformation and stress distribution of turbine parts is of great significance to ensure the safety, reliability, and service life of the system. Traditionally, the finite element analysis method has long dominated engineering simulation, but it often takes a lot of computing time and high computing costs when facing high-resolution three-dimensional models or complex load conditions. In addition, the finite element method relies on a complex meshing process and often requires a lot of manual intervention, which makes it limited in dealing with complex geometric structures. In addition, traditional methods are difficult to utilize the potential laws in a large amount of experimental data, which limits the ability to predict and optimize the design in real time. In recent years, the development of artificial intelligence, especially neural network methods, has brought new ideas to engineering simulation. These methods provide more efficient prediction and optimization methods by learning the approximate mapping relationship of complex physical systems. As an emerging model, Fourier neural operator (FNO) uses the powerful information processing capability of Fourier space to achieve fast prediction through nonlinear mapping, showing strong applicability. However, how to generate high-quality training data and how to handle complex geometric models remain challenges, especially in the field of solid mechanics, where there are relatively few applications and the elastic analysis of turbine parts has not yet fully explored its potential.

[0003] The Geo part excels in modeling complex geometries, such as point clouds, meshes, and the geometric structures of turbine parts, and has significant advantages in describing geometric features in complex physical systems. In the analysis of turbine parts, Geo is able to handle complex geometric shapes such as turbine blades and rotors, helping to accurately describe their physical properties. However, when faced with high dimensions or extremely complex geometric structures, Geo may require a lot of computing resources, and when dealing with complex geometric problems, it relies on a lot of domain knowledge and prior assumptions to describe geometric features and boundary conditions, which may become a computing bottleneck due to the diversity and complexity of turbine parts.

[0004] The Fourier Neural Operator (FNO) can efficiently solve partial differential equations. Especially when dealing with the elastic analysis of turbine parts, FNO can effectively capture complex physical phenomena such as thermal-mechanical coupling effects and stress distribution caused by rotation. FNO can accelerate calculations and improve accuracy by nonlinearly mapping data in Fourier space. Compared with traditional finite element methods, FNO does not rely on an explicit meshing process, can directly learn physical laws from data, and flexibly handle the complex geometries of turbine parts. Although FNO performs well in high-dimensional data processing, its reliance on high-quality training data means that the model may be limited when data is scarce or unknown. In addition, the "black box" nature of FNO may lead to poor interpretability of the model, especially in the elastic analysis of turbine parts, which may encounter certain challenges when strict verification and physical interpretation are required.

[0005] The combination of Geo-FNO can simultaneously utilize geometric information and frequency domain features, thereby improving the processing capabilities of elastic analysis of turbine parts. When analyzing turbine parts, Geo-FNO can not only accurately understand the impact of the spatial structure of turbine parts on stress and deformation, but also extract key features from the frequency domain. Through Fourier transform, Geo-FNO can reduce the need for direct processing of high-dimensional space, without the need for traditional gridding operations, making the solution of complex turbine parts more efficient, especially when faced with large-scale, diverse turbine parts of different shapes and physical conditions. The combination of Geo and FNO can significantly improve the calculation speed and accuracy. Summary of the invention

[0006] In response to the stress and strain prediction problem of complex structures, the present invention provides a virtual-reality interaction method for the elastic digital twin of turbine parts that combines geometric features with Fourier neural operators. It aims to combine finite element software and AI prediction, and use customized Fourier transform and geometric coding technology to improve the computational complexity of existing turbine part analysis methods and effectively handle complex geometric features. It optimizes traditional elastic analysis methods, reduces computational complexity, and improves analysis accuracy. The technology is suitable for solving complex geometric shapes and physical phenomena.

[0007] The objective of the present invention is achieved through the following technical solutions:

[0008] A virtual-real interaction method for elastic digital twins of turbine parts combining geometric features with Fourier neural operators comprises the following steps:

[0009] Step S1: Material properties and boundary conditions settings:

[0010] Use finite element analysis software to model turbine parts, and select key design parameters based on their actual working environment and conditions, including geometric parameters (such as blade shape and thickness), material property parameters (such as elastic modulus, Poisson's ratio, etc.), and loads and boundary conditions (such as the location and constraints of force application);

[0011] Step S2: Finite element data generation and format conversion:

[0012] Step S2.1, generating stress cloud diagrams of turbine parts before and after deformation by finite element analysis;

[0013] Step S2.2, extracting the coordinate information of the turbine part nodes and their corresponding stress and strain data, and exporting the data into .csv format, and further converting these data into .npy format through a customized script;

[0014] Step S3: Construction and training of prediction model based on Geometry-aware Fourier Operator (Geo-FNO):

[0015] The data generated in step S2 is preprocessed by geometric feature extraction and Fourier transform, and a Geo-FNO (Geometry-aware Fourier Neural Operator) prediction model is constructed and trained and optimized so that the model can accurately predict the stress distribution and elastic deformation of turbine parts under complex load conditions. The specific steps are as follows:

[0016] Step S3.1, data preprocessing:

[0017] Step S3.1.1, data standardization and dimensionality reduction: First, perform data standardization to eliminate the impact of physical quantities of different scales on model training and ensure that the data is at the same order of magnitude; then, use fast Fourier transform (FFT) to reduce the dimensionality of the node data of turbine parts, and extract the main frequency components of the node data through FFT. The Fourier transform formula is as follows:

[0018]

[0019] Where v(x) is a function defined in the physical space D, which represents the displacement field and stress field of the turbine parts; ψ(x,k) = e 2iπ＜x,k＞ is a standard Fourier basis, where k is a frequency domain variable and x is a point in physical space; μ(x) is a weight function; the integral represents the weighted sum of all points in physical space, and the transformed value in the frequency domain is calculated by the inner product with the basis function; the approximate sum In actual calculations, the physical space is discretized into a grid T consisting of a finite number of sampling points;

[0020] Step S3.1.2, geometric feature extraction and supplementation: The geometric features of the turbine parts are extracted by geometric coding technology and converted into feature vectors in high-dimensional space. The feature vectors not only contain the shape and size of the turbine parts, but also supplement the geometric information that may be omitted by Fourier transform through geometric transformation. The geometric feature mapping is:

[0021] X={x 1 ,x 2 ,x 3 ,...,x n}

[0022] In the formula, x 1 ,x 2 ,x 3 ,...,x n Represent key geometric features of turbine parts;

[0023] Step S3.2, geometry and computational space mapping:

[0024] Through coordinate transformation, the points in the physical space are mapped to the computational space, and Fourier transform processing is performed in the computational space, that is, by transforming φ a The physical space D a The points are mapped to D c Evaluate points in space:

[0025]

[0026] Where D c =[0,1] d is the normalized computational space;

[0027] Step S3.3, numerical solution:

[0028] The deformation behavior of turbine parts under complex load conditions is described by numerically solving equations. The numerical solution equation formula is:

[0029]

[0030] Where K is the stiffness matrix, which contains parameters related to the geometry and material properties of the turbine parts, v represents the vector field of node displacement and deformation, and v t represents the stress field, x represents the spatial coordinate, Fv t is the external load applied to the model, F represents the load intensity, represents the scaling factor;

[0031] Step S3.4, frequency domain restoration:

[0032] Through inverse Fourier transform, the frequency domain data is converted into functions in physical space, and the stress and deformation physical quantities of turbine parts are restored;

[0033] Step S3.5, model training:

[0034] Train the Geo-FNO-based prediction model to accurately predict the stress distribution and elastic deformation of turbine parts under complex loading conditions;

[0035] Step S4: Model prediction and verification:

[0036] The test data is input into the trained Geo-FNO prediction model to quickly predict the stress distribution and elastic deformation of turbine parts. Subsequently, the prediction accuracy and reliability of the Geo-FNO prediction model are verified by comparing the stress values ​​and deformations in the finite element analysis calculation results.

[0037] Compared with the prior art, the present invention has the following advantages:

[0038] 1. Improvement of computational efficiency: Through customized Fourier transform and geometric coding technology, the computational time of the present invention is reduced by 50-80% compared with the traditional method.

[0039] 2. Ability to adapt to complex geometric shapes: For parts with complex geometric shapes (such as turbine blades), traditional methods require a lot of meshing and recalculation, while the geometric encoding technology of the present invention can complete mesh adaptation and deformation processing in 30 to 50% of the time, greatly improving the adaptability to complex geometric shapes.

[0040] 3. Resource utilization efficiency: Through Fourier transform and data dimension reduction processing, the present invention reduces the demand for memory, especially when processing large-scale data.

[0041] 4. The present invention maps the physical space into a regular computing space, characterizes the high-dimensional space in combination with geometric features, uses Fourier neural operators for efficient frequency domain analysis, optimizes the node data processing flow of turbine parts, and significantly reduces the computational complexity.

[0042] 5. Compared with traditional elastic analysis methods, the method of the present invention greatly improves the calculation speed and resource utilization efficiency while ensuring accuracy, especially when dealing with complex geometric structures.

[0043] 6. The present invention has broad application prospects, especially in engineering analysis in the fields of turbomachinery, aerospace, etc., and can provide efficient computing support for design optimization, performance evaluation and fault detection. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 This is the prediction result diagram;

[0045] Figure 2This is the actual result diagram;

[0046] Figure 3 It is a difference graph. DETAILED DESCRIPTION

[0047] The technical solution of the present invention is further described below in conjunction with the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be included in the protection scope of the present invention.

[0048] The present invention provides a virtual-real interaction method for elastic digital twins of turbine parts combining geometric features with Fourier neural operators, the method comprising the following steps:

[0049] Step S1: Material properties and boundary condition settings:

[0050] Finite element analysis software is used to model turbine parts, and key design parameters are selected based on their actual working environment and conditions, including geometric parameters (such as blade shape and thickness), material property parameters (such as elastic modulus, Poisson's ratio, etc.), and loads and boundary conditions (such as the location of force application and constraints).

[0051] In this step, the design parameters are reasonably selected according to the actual working environment and working conditions of the turbine parts to ensure that the analysis results can truly reflect the mechanical behavior of the turbine parts under working conditions. The key design parameters of the turbine parts include:

[0052] (1) Geometric shape parameters, generated by 3D modeling tools or imported into CAD files in standard formats;

[0053] (2) Material property parameters, including elastic modulus, Poisson’s ratio, and density, are set based on the actual use conditions of the part;

[0054] (3) Load boundary conditions, which are set based on the actual working environment and include fixed constraints and loads.

[0055] Step S2: Finite element data generation and format conversion:

[0056] The stress cloud diagrams of the turbine parts before and after deformation are generated by finite element analysis; the coordinate information of the nodes of the turbine parts and their corresponding stress and strain data are extracted, and the data is exported to .csv format, which is further converted to .npy format through a custom script for subsequent Geo-FNO model processing and analysis. This step ensures the standardization of the data format and processing efficiency.

[0057] In this step, the coordinate information of the turbine part nodes and the corresponding stress and strain data are extracted through the field output interface. The generated data is adapted to the format requirements of the Geo-FNO model input. The specific steps are as follows:

[0058] (1) Generate stress tensor and strain tensor data at the nodes using the post-processing module of the finite element analysis software;

[0059] (2) According to the topological structure of the finite element model, the global coordinates of the nodes are extracted and mapped to the grid elements to ensure the consistency and accuracy of all data points in the computational domain;

[0060] (3) Through a custom script, the extracted data is exported in .csv format and further converted into .npy format to adapt it to the input requirements of the Geo-FNO model. This data format conversion process effectively improves the efficiency and accuracy of model training.

[0061] Step S3: Geo-FNO prediction model construction and training:

[0062] In this step, the data generated in step S2 are preprocessed by geometric feature extraction and Fourier transform to construct a Geo-FNO prediction model, which is then trained and optimized so that the model can accurately predict the stress distribution and elastic deformation of turbine parts under complex load conditions.

[0063] 1. Data preprocessing:

[0064] (1) Data standardization and dimensionality reduction: In order to improve the computational efficiency of the model and ensure that complex geometric features and physical properties can be effectively retained, data preprocessing is first standardized. The purpose of standardization is to eliminate the impact of physical quantities of different scales on model training and ensure that the data is at the same order of magnitude. Next, the fast Fourier transform (FFT) is used to reduce the dimensionality of the node data of the turbine parts. The main frequency components of the node data are extracted by FFT to reduce the computational complexity and improve data processing efficiency. The Fourier transform process can map the physical information of the high-dimensional space to the frequency domain space, while ensuring that the reduced-dimensional data can still fully represent the geometric features and physical properties. The Fourier transform formula is as follows:

[0065]

[0066] Formula (1) is a Fourier transform formula combined with geometric coding technology. v(x) is a function defined in the physical space D, which represents the displacement field and stress field of the turbine parts; ψ(x, k) = e 2iπ＜x,k＞is a standard Fourier basis, where k is a frequency domain variable, the exponent represents different frequency components, and x is a point in physical space; μ(x) is a weight function that indicates the importance of each point; the integral represents the weighted sum of all points in physical space, and the transformed value in the frequency domain is calculated by the inner product with the basis function; the approximate sum In actual calculations, the physical space is usually discretized into a grid T consisting of a finite number of sampling points, so the actual Fourier transform is approximated by the integration of discrete points in the grid. In this way, Geo-FNO can combine geometric information with frequency domain analysis to improve analysis accuracy and accelerate the calculation process.

[0067] (2) Geometric feature extraction and supplementation: In order to ensure that the geometric information of turbine parts can be fully reflected in the data after dimensionality reduction, the geometric features of turbine parts are extracted through geometric coding technology and converted into feature vectors in high-dimensional space. This feature vector not only contains the shape and size of the turbine parts, but also supplements the geometric information that may be missed by Fourier transform through geometric transformation, such as width, height, volume, angle, radius, etc. Specifically, geometric coding technology captures the details of geometric features in physical space by constructing basis functions suitable for the geometric shape, making its representation in high-dimensional space more comprehensive and providing more accurate input for subsequent model training. The geometric feature mapping is:

[0068] X={x 1 ,x 2 ,x 3 ,...,x n}(2)

[0069] In formula (2), x 1 ,x 2 ,x 3 ,...,x n The key geometric features of turbine parts can be represented, and these features can be transformed into feature vectors in high-dimensional space through parametric mapping.

[0070] 2. Geometry and computational space mapping:

[0071] In order to ensure that the geometric features in the physical space can adapt to the Fourier transform, a differentiable coordinate transformation is constructed to map the geometric features of the physical space to the regular computational space. Specifically, through the coordinate transformation, the points in the physical space are mapped to the computational space, and the Fourier transform processing is performed in the computational space. This process ensures the consistency of the geometric features in the Fourier transform through effective coordinate transformation, and retains the structural information of the physical space, so that the subsequent frequency domain analysis can more accurately reflect the geometric characteristics of the turbine parts. Specifically, the goal is to find a smooth homeomorphic transformation between the physical space and the computational space, that is, by transforming φ a The physical space Da The points are mapped to D c Evaluate points in space:

[0072]

[0073] In formula (3), D c =[0,1] d It is the standardized computational space. Through this transformation, the complex geometry of the physical space is mapped to a uniform grid in the computational space.

[0074] 3. Numerical solution:

[0075] Numerical solution equation: In the elastic analysis process of turbine parts, it is necessary to establish a numerical model that can accurately predict stress, strain and deformation. In the present invention, the core of model construction is to describe the deformation behavior of turbine parts under complex load conditions by numerical solution equation. This equation is the equation of the numerical solution process, and the formula is:

[0076]

[0077] In formula (4), K is the stiffness matrix, which contains parameters related to the geometry and material properties of the turbine parts, v represents the vector field of node displacement and deformation, and v t Represents the stress field, x represents the spatial coordinate, and the right half Fv t is the external load applied to the model, where F represents the load intensity. Represents the scaling factor, which is used to adjust the response of the system. In general, by solving this equation, the stress distribution and elastic deformation prediction of turbine parts under complex load conditions can be achieved, providing a basis for subsequent model training.

[0078] 4. Frequency domain restoration:

[0079] After Fourier transform, the frequency domain data needs to be restored to the physical space in order to be used for actual physical quantity analysis. Through the inverse Fourier transform, the frequency domain data is converted into a function in the physical space to restore the physical quantities such as stress and deformation of the turbine parts. This process ensures the consistency and integrity between the geometric features and the frequency domain features, so that the frequency domain analysis results can correspond to the actual mechanical behavior in the physical space. The formula for the inverse Fourier transform is as follows:

[0080]

[0081] In formula (5), is the Fourier specification in the frequency domain, e ikx is a complex exponential kernel, representing the inverse mapping from frequency domain to space.

[0082] 5. Model training:

[0083] The Geo-FNO-based prediction model is trained to enable it to accurately predict the stress distribution and elastic deformation of turbine parts under complex loading conditions.

[0084] Step S4: Model prediction and verification:

[0085] The test data is input into the trained Geo-FNO prediction model to quickly predict the stress distribution and elastic deformation of turbine parts. Subsequently, the prediction accuracy and reliability of the Geo-FNO prediction model are verified by comparing the stress values ​​and deformations in the finite element analysis calculation results.

[0086] In this step, by comparing the finite element calculation results and the Geo-FNO prediction results, the Adam optimizer and mean square error (MSE) are used to evaluate the model prediction performance, and the model is optimized in a targeted manner, where:

[0087] (1) Optimizer: The Adam optimizer combines the ideas of momentum and adaptive adjustment and can effectively handle complex nonlinear relationships. It adaptively adjusts the learning rate of each parameter by maintaining the first-order moment (i.e., the mean of the gradient) and the second-order moment (i.e., the variance of the gradient), thereby accelerating convergence and improving stability. The specific steps are as follows: In each iteration, Adam calculates the mean and variance of the current gradient and uses this information to adjust the update step size of each parameter; by introducing momentum, the Adam optimizer can speed up convergence and avoid slow convergence caused by small gradients or oscillations during training; when dealing with stress prediction tasks for complex physical systems such as turbine parts, the Adam optimizer usually performs better than the traditional gradient descent method.

[0088] (2) Mean square error (MSE) evaluation index: Mean square error is a commonly used regression model evaluation index, which is used to measure the difference between the model prediction value and the true value. In this step, MSE is used to quantify the difference between the Geo-FNO model prediction results and the finite element calculation results. Its calculation formula is:

[0089]

[0090] In formula (6), y pred (i) is the predicted value of the i-th sample, y true (i) is the true value of the i-th sample, and n is the total number of samples.

[0091] (3) Optimization process: During the training process, the Geo-FNO model gradually reduces the error between the prediction results and the real data through iterative optimization. The Adam optimizer adaptively adjusts the update step size of each parameter, allowing the model to effectively learn the nonlinear relationship in the data. In order to optimize the model, MSE is used as the loss function to minimize the loss value in each iteration, thereby improving the prediction accuracy of the model. The optimization of the loss function directly reflects the degree of fit of the model on the training data.

[0092] (4) Result visualization: The visualization of the comparison results shows the difference between the prediction and the actual data by superimposing the stress cloud map, so as to intuitively verify the applicability of the Geo-FNO model.

[0093] Based on Geo-FNO, an improved version of FNO, the present invention combines the engineering characteristics of turbine parts, and through efficient model training and prediction processes, the elastic analysis method based on Geo-FNO can quickly and accurately predict the stress and deformation distribution of turbine parts. This method can significantly reduce the computational cost while accurately predicting the stress and deformation distribution of turbine parts under complex load conditions, providing a new efficient and low-cost new technology solution for the design optimization, real-time health monitoring and fault diagnosis of turbine parts.

[0094] Example:

[0095] This embodiment provides a virtual-real interaction method for elastic digital twins of turbine parts combining geometric features with Fourier neural operators, and the method comprises the following steps:

[0096] Step 1: Material properties and boundary conditions settings:

[0097] A local flat area on the blade surface is intercepted, and the actual geometric features of the blade are simplified into a plane model. According to the material properties and loading conditions of the actual blade, the density is 6300, the Young's modulus is 210000000, and the Poisson's ratio is 0.3. In this area, a force from top to bottom is given. This force is used for the impact force of water flow in actual working conditions. It is mainly used to analyze local fatigue and analyze fatigue problems that may occur in high stress concentration areas. (This embodiment focuses on solid mechanics, that is, the blade part, not dynamics.)

[0098] Step 2: Geo-FNO prediction model training:

[0099] The coordinate data and stress-strain data are divided into three independent data files, and the stress-strain data are normalized to construct a regular grid. Normalization helps to reduce the impact of data of different scales on model training and ensure that the contribution of each feature to the model is close. In this embodiment, a 64×64 grid is used as an example to interpolate strain and stress data into a regular grid and convert it into input and output tensors. The cubic method is used for interpolation to smooth the data and better fit possible nonlinear distributions. Fill_value=0 is used to fill empty values ​​in order to avoid interference of missing values ​​after interpolation on model training. The hyperparameters in this embodiment are set to batchsize=1, modes1=modes2=16, width=64, epochs=1000, and lr=0.001. Due to the small amount of data, small batch training is used to maintain high training accuracy or improve computational efficiency; the first Fourier mode number determines the complexity of the frequency domain representation; the second Fourier mode number ensures that the model can capture more features.

[0100] Step 3: Geo-FNO prediction model evaluation:

[0101] The Adam optimizer and MSE loss function are used for training. The Adam optimizer combines momentum and adaptive adjustment, and performs well in dealing with nonlinear relationships. The training speed is fast and stable. The MSE loss function can effectively measure the prediction error of the model, and is especially suitable for dealing with continuous value regression problems, such as predicting physical quantities such as stress and strain. Finally, the heat map of each sample is visualized and saved. The prediction results are shown in the figure below. Figure 1 The actual result is shown in the figure Figure 2 As shown, the difference diagram is Figure 3 shown.

[0102] Figure 1 and Figure 2 In the figure, the X-axis and Y-axis represent the spatial coordinates of the object, the X-axis is the horizontal coordinate, and the Y-axis is the vertical coordinate. It represents the specific position of the turbine part on the two-dimensional plane. The color bar on the right represents the intensity of the stress value, ranging from 0 to 1. The closer the color is to yellow, the stronger the stress is, and the closer to purple, the weaker the stress is. From the color gradient in the figure, it can be seen that the stress in the upper area is larger and the stress in the lower area is smaller. This shows that in the upper part of the image, in the square area cut by the blade, the upper half is a stress concentration area, corresponding to this area of ​​the component that is subjected to high stress intensity.

[0103] Figure 3In the figure, the X-axis and Y-axis represent the spatial coordinates of the data; the color bar represents the range of values ​​that have no corresponding color, and the colors range from purple (indicating negative values) to yellow (indicating positive values), indicating that the difference in the surface prediction can change from negative to positive, that is, the accuracy of the prediction. In the lower right area of ​​the figure, there are more purples, indicating that the predicted value is lower than the actual value, that is, the model underestimates the result, and the performance needs to be improved. Most of it is green, indicating that the prediction is accurate. The yellow part indicates that the predicted result is higher than the actual value, and the model overestimates the result.

Claims

1. A virtual-real interaction method for elastic digital twins of turbine parts combining geometric features and Fourier neural operators, characterized in that The method comprises the following steps: Step S1: Material properties and boundary condition settings: Use finite element analysis software to model turbine parts and select key design parameters, including geometric shape parameters, material property parameters, loads and boundary conditions, based on their actual working environment and conditions; Step S2: Finite element data generation and format conversion: Step S2.1, generating stress cloud diagrams of turbine parts before and after deformation by finite element analysis; Step S2.2, extracting the coordinate information of the turbine part nodes and their corresponding stress and strain data, and exporting the data into .csv format, and further converting these data into .npy format through a customized script; Step S3: Construction and training of prediction model based on geometric perception Fourier operator: The data generated in step S2 is preprocessed by geometric feature extraction and Fourier transform, a Geo-FNO prediction model is constructed, and the model is trained and optimized so that the model can accurately predict the stress distribution and elastic deformation of turbine parts under complex load conditions; Step S4: Model prediction and verification: The test data is input into the trained Geo-FNO prediction model to quickly predict the stress distribution and elastic deformation of turbine parts. Subsequently, the prediction accuracy and reliability of the Geo-FNO prediction model are verified by comparing the stress values ​​and deformations in the finite element analysis calculation results.

2. The virtual-real interaction method of elastic digital twin of turbine parts combining geometric features and Fourier neural operators according to claim 1 is characterized in that In step S2, the coordinate information of the turbine part nodes and the corresponding stress and strain data are extracted through the field output interface, and the generated data is adapted to the format requirements of the Geo-FNO model input. The specific steps are as follows: (1) Generate stress tensor and strain tensor data at the nodes using the post-processing module of the finite element analysis software; (2) According to the topological structure of the finite element model, the global coordinates of the nodes are extracted and mapped to the grid elements to ensure the consistency and accuracy of all data points in the computational domain; (3) Through a custom script, the extracted data is exported in .csv format and further converted into .npy format to adapt it to the Geo-FNO model input requirements.

3. The virtual-real interaction method of elastic digital twin of turbine parts combining geometric features and Fourier neural operator according to claim 1 is characterized in that The specific steps of step S3 are as follows: Step S3.1, data preprocessing: Step S3.1.1, data standardization and dimensionality reduction: First, perform data standardization to eliminate the influence of physical quantities of different scales on model training and ensure that the data are at the same order of magnitude; then, use fast Fourier transform (FFT) to reduce the dimensionality of the node data of turbine parts and extract the main frequency components of the node data through FFT; Step S3.1.2, geometric feature extraction and supplementation: The geometric features of the turbine parts are extracted by geometric coding technology and converted into feature vectors in high-dimensional space. The feature vectors not only contain the shape and size of the turbine parts, but also supplement the geometric information that may be missed by Fourier transform through geometric transformation; Step S3.2, Geometry and computational space mapping: Through coordinate transformation, the points in the physical space are mapped to the computational space, and Fourier transform processing is performed in the computational space; Step S3.3, numerical solution: The deformation behavior of turbine parts under complex load conditions is described by numerically solving equations. The numerical solution equation formula is: Where K is the stiffness matrix, which contains parameters related to the geometry and material properties of the turbine parts, v represents the vector field of node displacement and deformation, and v t represents the stress field, x represents the spatial coordinate, Fv t is the external load applied to the model, F represents the load intensity, represents the scaling factor, and D represents the physical space; Step S3.4, frequency domain restoration: Through inverse Fourier transform, the frequency domain data is converted into functions in physical space, and the stress and deformation physical quantities of turbine parts are restored; Step S3.5, model training: The Geo-FNO-based prediction model is trained to enable it to accurately predict the stress distribution and elastic deformation of turbine parts under complex loading conditions.

4. The virtual-real interaction method of elastic digital twin of turbine parts combining geometric features and Fourier neural operators according to claim 1 is characterized in that In step S3.1.1, the Fourier transform formula is as follows: Where v(x) is a function defined in the physical space D, which represents the displacement field and stress field of the turbine parts; ψ(x,k) = e 2i π＜x,k＞ is a standard Fourier basis, where k is a frequency domain variable and x is a point in physical space; μ(x) is a weight function; the integral represents the weighted sum of all points in physical space, and the transformed value in the frequency domain is calculated by the inner product with the basis function; the approximate sum It means that in actual calculation, the physical space is discretized into a grid T consisting of a finite number of sampling points.

5. The virtual-real interaction method of elastic digital twin of turbine parts combining geometric features and Fourier neural operator according to claim 1 is characterized in that In step S3.1.2, the geometric feature mapping is: X={x1,x2,x3,...,x n } In the formula, x1,x2,x3,...,x n Represents the key geometric features of a turbine part.

6. The virtual-real interaction method of elastic digital twin of turbine parts combining geometric features and Fourier neural operator according to claim 1 is characterized in that In step S3.2, by transforming φ a The physical space D a The points are mapped to D c Evaluate points in space: Where D c =[0,1] d is the normalized computational space.

7. The virtual-real interaction method for elastic digital twins of turbine parts combining geometric features and Fourier neural operators according to claim 1 is characterized in that In step S4, the Adam optimizer and mean square error are used to evaluate the prediction performance of the model, and the model is optimized in a targeted manner.

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