A turbine part elastic digital twin virtual-real interaction method combining geometric features and fourier neural operator

Abstract

This invention discloses a method for interactive virtual-real digital twins of turbine components, combining geometric features and Fourier neural operators. The method includes the following steps: Step S1, setting material properties and boundary conditions; Step S2, generating and converting finite element data; Step S3, constructing and training a prediction model based on geometrically aware Fourier operators; Step S4, model prediction and verification. This invention utilizes a combination of finite element software and AI prediction to improve the computational complexity of existing turbine component analysis methods and effectively handle complex geometric features through custom Fourier transforms and geometric encoding techniques. It optimizes traditional elastic analysis methods, reducing computational complexity while improving analysis accuracy. This technique is applicable to solving complex geometries and physical phenomena. This invention has broad application prospects and can provide efficient computational support for design optimization, performance evaluation, and fault detection.

Description

Technical Field

[0001] This invention belongs to the field of intelligent simulation and relates to a method for virtual-real interaction of elastic digital twins of turbine parts, specifically a method for virtual-real interaction of elastic digital twins of turbine parts that combines geometric features and Fourier neural operators. Background Technology

[0002] Turbine components are critical parts of modern mechanical systems, widely used in high-tech fields such as aero-engines, gas turbines, and wind turbines. These turbine components typically face extreme operating 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 components is crucial for ensuring the safety, reliability, and service life of the system. Traditionally, the finite element method (FEM) has dominated engineering simulation, but it often requires significant computational time and costs when dealing with high-resolution 3D models or complex load conditions. Furthermore, the FEM relies on a complex mesh generation process, often requiring substantial manual intervention, which limits its ability to handle complex geometries. Additionally, traditional methods struggle to utilize the latent patterns in large amounts of experimental data, limiting their real-time prediction and optimization capabilities. 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 tools by learning approximate mapping relationships in complex physical systems. The Fourier Neural Operator (FNO), as an emerging model, utilizes the powerful information processing capabilities of Fourier space to achieve rapid prediction through nonlinear mapping, demonstrating strong applicability. However, generating high-quality training data and handling complex geometric models remain challenges, especially given the relatively limited applications in solid mechanics and the untapped potential of elastic analysis of turbine components.

[0003] The Geo component excels at modeling complex geometries, such as point clouds, meshes, and the geometry of turbine components, demonstrating significant advantages in describing the geometric features of complex physical systems. In turbine component analysis, Geo can handle complex geometries such as turbine blades and rotors, helping to accurately describe their physical properties. However, when dealing with high-dimensional or extremely complex geometries, Geo may require substantial computational resources. Furthermore, when handling complex geometric problems, it relies heavily on domain knowledge and prior assumptions to describe geometric features and boundary conditions, which can become a computational bottleneck given the diversity and complexity of turbine components.

[0004] Fourier Neural Operators (FNOs) are highly efficient at solving partial differential equations, particularly in the elastic analysis of turbine components. FNOs can effectively capture complex physical phenomena, such as thermo-mechanical coupling effects and stress distribution caused by rotation. By performing nonlinear mapping of data in Fourier space, FNOs accelerate computation and improve accuracy. Compared to traditional finite element methods, FNOs do not rely on explicit meshing processes and can directly learn physical laws from the data, flexibly handling the complex geometries of turbine components. While FNOs excel in high-dimensional data processing, their dependence on high-quality training data means that the model may be limited when data is scarce or unknown. Furthermore, the "black box" nature of FNOs can lead to poor model interpretability, especially in the elastic analysis of turbine components, where rigorous verification and physical interpretation may be challenging.

[0005] The combination of Geo-FNO and FNO can simultaneously utilize geometric information and frequency domain features, thereby enhancing the processing capabilities for elastic analysis of turbine components. When analyzing turbine components, Geo-FNO can not only accurately understand the influence of the spatial structure of the turbine component 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 spaces, eliminating the need for traditional meshing operations, making the solution of complex turbine components more efficient. Especially when dealing with large-scale turbine components with diverse shapes and physical conditions, combining Geo-FNO can significantly improve computational speed and accuracy. Summary of the Invention

[0006] To address the problem of stress-strain prediction for complex structures, this invention provides a virtual-real interaction method for elastic digital twins of turbine components that combines geometric features with Fourier neural operators. This method aims to improve upon existing turbine component analysis methods by combining finite element software and AI prediction, using customized Fourier transforms and geometric coding techniques to effectively handle complex geometric features. It optimizes traditional elastic analysis methods, reducing computational complexity while improving analytical accuracy. This technique is applicable to solving complex geometries and physical phenomena.

[0007] The objective of this invention is achieved through the following technical solution:

[0008] A method for interactive virtual-real relationships in the elastic digital twin of turbine components, combining geometric features and Fourier neural operators, includes the following steps:

[0009] Step S1: Setting material properties and boundary conditions:

[0010] The turbine parts were modeled using finite element analysis software, and key design parameters were 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 and Poisson's ratio), and loads and boundary conditions (such as the location of applied forces and constraint conditions).

[0011] Step S2, Limited Metadata Generation and Format Conversion:

[0012] Step S2.1: Generate stress cloud diagrams of the turbine parts before and after deformation through finite element analysis;

[0013] Step S2.2: Extract the coordinate information of the turbine component nodes and their corresponding stress and strain data, and export the data as .csv format. Then, use a custom script to further convert these data into .npy format.

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

[0015] The data generated in step S2 is preprocessed by geometric feature extraction and Fourier transform to construct a Geo-FNO (Geometry-aware Fourier Neural Operator) prediction model, which is then trained and optimized to enable the model to accurately predict the stress distribution and elastic deformation of turbine components 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, data standardization is performed to eliminate the influence of physical quantities at different scales on model training, ensuring that the data are of the same order of magnitude. Next, Fast Fourier Transform (FFT) is used to reduce the dimensionality of the turbine component's nodal data. The main frequency components of the nodal data are extracted using FFT. The Fourier Transform formula is as follows:

[0018]

[0019] In the formula, v(x) is a function defined in physical space D, representing the displacement field and stress field of the turbine component; ψ(x,k)=e 2iπ＜x,k＞ It is a standard Fourier basis, where k is a frequency domain variable, x is a point in physical space; μ(x) is a weighting function; the integral represents a weighted summation over all points in physical space, and the transform value in the frequency domain is calculated by the inner product with the basis functions; approximate summation. This means that 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: Geometric features of the turbine parts are extracted using geometric encoding technology and transformed into feature vectors in a high-dimensional space. These feature vectors not only contain the shape and dimensions of the turbine parts but also supplement geometric information that might have been missed by the Fourier transform through geometric transformation. The geometric feature mapping is as follows:

[0021] X = {x1, x2, x3, ..., x} n}

[0022] In the formula, x1, x2, x3, ..., x n This indicates the key geometric features of the turbine components;

[0023] Step S3.2, Geometric and Computational Space Mapping:

[0024] By transforming coordinates, points in the physical space are mapped to the computational space, and then Fourier transforms are performed in the computational space, i.e., by transforming φ a physical space D a The points are mapped to D c Points in computational space:

[0025]

[0026] In the formula, D c =[0,1] d It is the standardized computational space;

[0027] Step S3.3, Numerical Solution:

[0028] The deformation behavior of turbine components under complex load conditions is described by numerically solving the equations. The numerical solution equations are as follows:

[0029]

[0030] In the formula, K is the stiffness matrix, containing parameters related to the geometry and material properties of the turbine components, and v represents the vector field of displacement and deformation of the nodes. t Represents the stress field, x represents the spatial coordinate, Fv t It is the external load applied to the model, where F represents the load intensity. Indicates the scaling factor;

[0031] Step S3.4, Frequency Domain Reconstruction:

[0032] By using inverse Fourier transform, the frequency domain data is converted into a function in physical space, and the stress and deformation physical quantities of the turbine parts are recovered.

[0033] Step S3.5, Model Training:

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

[0035] Step S4, Model Prediction and Validation:

[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. Then, the prediction accuracy and reliability of the Geo-FNO prediction model are verified by comparing the stress values ​​and deformation amounts in the finite element analysis calculation results.

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

[0038] 1. Improved computational efficiency: By using custom Fourier transform and geometric coding techniques, the computation time of this invention is reduced by 50-80% compared to traditional methods.

[0039] 2. Ability to adapt to complex geometries: For parts with complex geometries (such as turbine blades), traditional methods require a large amount of mesh generation and recalculation, while the geometric coding technology of this invention can complete mesh adaptation and deformation processing in 30 to 50% of the time, which greatly improves the adaptability to complex geometries.

[0040] 3. Resource utilization efficiency: Through Fourier transform and data dimensionality reduction, this invention reduces the memory requirement, especially when processing large-scale data.

[0041] 4. This invention maps physical space to a regular computational space, combines geometric features to characterize high-dimensional space, and uses Fourier neural operators for efficient frequency domain analysis, thereby optimizing the node data processing flow of turbine parts and significantly reducing computational complexity.

[0042] 5. Compared with traditional elasticity analysis methods, the method of this invention significantly improves calculation speed and resource utilization efficiency while ensuring accuracy, and has obvious advantages, especially when dealing with complex geometric structures.

[0043] 6. This invention has broad application prospects, especially in engineering analysis in fields such as turbomachinery and aerospace, where it can provide efficient computational support for design optimization, performance evaluation, and fault detection. Attached Figure Description

[0044] Figure 1 The prediction results are shown in the image.

[0045] Figure 2 This is a diagram showing the actual results;

[0046] Figure 3 This is a difference graph. Detailed Implementation

[0047] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0048] This invention provides a method for interactive virtual-real digital twins of turbine components that combines geometric features with Fourier neural operators. The method includes the following steps:

[0049] Step S1: Setting material properties and boundary conditions:

[0050] The turbine components were modeled using finite element analysis software, and key design parameters were selected based on their actual working environment and conditions. These parameters included geometric parameters (such as blade shape and thickness), material property parameters (such as elastic modulus and Poisson's ratio), and loads and boundary conditions (such as the location of applied forces and constraint conditions).

[0051] In this step, design parameters are rationally selected based on the actual working environment and conditions of the turbine components to ensure that the analysis results accurately reflect the mechanical behavior of the turbine components under operating conditions. The key design parameters of the turbine components include:

[0052] (1) Geometric parameters are generated by 3D modeling tools or imported into standard format CAD files;

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

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

[0055] Step S2, Limited Metadata Generation and Format Conversion:

[0056] Stress cloud diagrams of the turbine component before and after deformation were generated using finite element analysis. The coordinate information of the turbine component nodes and their corresponding stress and strain data were extracted and exported as a .csv file. A custom script then further converted this data to a .npy file for subsequent processing and analysis using the Geo-FNO model. This step ensured data format standardization and processing efficiency.

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

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

[0059] (2) Based on the topology of the finite element model, extract the global coordinates of the nodes and map them to the mesh cells to ensure the consistency and accuracy of all data points in the computational domain;

[0060] (3) The extracted data is exported in .csv format using a custom script and then further converted to .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 is 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] I. Data Preprocessing:

[0064] (1) Data Standardization and Dimensionality Reduction: To improve the computational efficiency of the model and ensure the effective preservation of complex geometric features and physical properties, standardization is performed first in data preprocessing. The purpose of standardization is to eliminate the influence of physical quantities at different scales on model training and ensure that the data are on the same order of magnitude. Next, Fast Fourier Transform (FFT) is used to reduce the dimensionality of the nodal data of the turbine parts. The main frequency components of the nodal data are extracted by FFT, reducing computational complexity and improving data processing efficiency. The Fourier Transform process can map physical information in high-dimensional space to frequency domain space, while ensuring that the dimensionality-reduced data can still fully represent geometric features and physical properties. The Fourier Transform formula is as follows:

[0065]

[0066] Formula (1) is a Fourier transform formula that incorporates geometric coding technology. v(x) is a function defined in physical space D, representing the displacement field and stress field of the turbine parts; ψ(x,k)=e 2iπ＜x,k＞ It 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 weighting function, indicating the importance of each point; the integral represents a weighted summation over all points in physical space, and the transform value in the frequency domain is calculated by the inner product with the basis functions; approximate summation. In practical calculations, the physical space is typically discretized into a grid T consisting of a finite number of sampling points. Therefore, the actual Fourier transform is approximated by integrating the discrete points within the grid. In this way, Geo-FNO can combine geometric information with frequency domain analysis, improving analytical accuracy and accelerating the computation process.

[0067] (2) Geometric Feature Extraction and Supplementation: To ensure that the geometric information of the turbine parts is fully represented in the dimensionality-reduced data, geometric encoding techniques are used to extract the geometric features of the turbine parts and transform them into feature vectors in a high-dimensional space. These feature vectors not only contain the shape and size of the turbine parts but also supplement geometric information that might be missed by the Fourier transform, such as width, height, volume, angle, and radius, through geometric transformation. Specifically, geometric encoding techniques construct basis functions suitable for the geometric shape to capture the details of geometric features in physical space, making its representation in high-dimensional space more comprehensive and providing more accurate input for subsequent model training. The geometric feature mapping is as follows:

[0068] X = {x1, x2, x3, ..., x} n}(2)

[0069] In formula (2), x1, x2, x3, ..., x n It can represent the key geometric features of turbine parts, which can be transformed into feature vectors in a high-dimensional space through parametric mapping.

[0070] II. Geometric and Computational Space Mapping:

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

[0072]

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

[0074] III. Numerical Solution:

[0075] Numerical Solution Equations: In the elastic analysis of turbine components, it is necessary to establish a numerical model capable of accurately predicting stress, strain, and deformation. In this invention, the core of model construction lies in describing the deformation behavior of turbine components under complex load conditions through numerical solution equations. These equations are the equations for the numerical solution process, and the formulas are as follows:

[0076]

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

[0078] IV. Frequency Domain Reconstruction:

[0079] After the Fourier transform, the frequency domain data needs to be restored to physical space for practical physical quantity analysis. The inverse Fourier transform converts the frequency domain data into functions in physical space, recovering physical quantities such as stress and deformation of the turbine components. This process ensures consistency and integrity between geometric and frequency domain features, allowing the frequency domain analysis results to correspond to actual mechanical behavior in physical space. The formula for the inverse Fourier transform is as follows:

[0080]

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

[0082] V. Model Training:

[0083] Train a Geo-FNO-based prediction model to accurately predict the stress distribution and elastic deformation of turbine components under complex load conditions.

[0084] Step S4, Model Prediction and Validation:

[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. Then, the prediction accuracy and reliability of the Geo-FNO prediction model are verified by comparing the stress values ​​and deformation amounts in the finite element analysis calculation results.

[0086] In this step, the model's predictive performance is evaluated by comparing the finite element analysis results and the Geo-FNO prediction results, using the Adam optimizer and mean squared error (MSE). The model is then optimized accordingly.

[0087] (1) Optimizer: The Adam optimizer combines momentum and adaptive adjustment to effectively handle complex nonlinear relationships. It accelerates convergence and improves stability by maintaining the first moment (mean of the gradient) and the second moment (variance of the gradient) and adaptively adjusting the learning rate of each parameter. 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 accelerate convergence and avoid slow convergence due to small gradients or oscillations during training; when dealing with stress prediction tasks of complex physical systems such as turbine parts, the Adam optimizer typically outperforms the traditional gradient descent method.

[0088] (2) Mean Squared Error (MSE) Evaluation Metric: Mean squared error is a commonly used evaluation metric for regression models, used to measure the difference between the model's predicted values ​​and the actual values. In this step, MSE is used to quantify the difference between the Geo-FNO model's prediction results and the finite element calculation results. Its calculation formula is as follows:

[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 training, the Geo-FNO model iteratively optimizes to gradually reduce the error between the predicted results and the actual data. The Adam optimizer adaptively adjusts the update step size of each parameter, enabling the model to effectively learn the nonlinear relationships in the data. To optimize the model, MSE is used as the loss function, minimizing the loss value in each iteration to improve the model's prediction accuracy. The optimization of the loss function directly reflects the degree of fit of the model to the training data.

[0092] (4) Results visualization: The results are visualized by overlaying stress cloud maps to show the difference between the prediction and the actual data, so as to intuitively verify the applicability of the Geo-FNO model.

[0093] This invention, based on Geo-FNO, an improved version of FNO, and considering the engineering characteristics of turbine components, utilizes an efficient model training and prediction process and Geo-FNO-based elastic analysis methods to quickly and accurately predict the stress and deformation distribution of turbine components. This method significantly reduces computational costs while accurately predicting the stress and deformation distribution of turbine components under complex load conditions, providing an efficient and low-cost new technological solution for turbine component design optimization, real-time health monitoring, and fault diagnosis.

[0094] Example:

[0095] This embodiment provides a method for interactive virtual-real relationships in the elastic digital twin of turbine components, combining geometric features with Fourier neural operators. The method includes the following steps:

[0096] Step 1: Setting Material Properties and Boundary Conditions

[0097] A locally flat area on the blade surface is selected, and the actual geometric features of the blade are simplified into a planar model based on the actual material properties and loading conditions of the blade. The density is 6300, Young's modulus is 210,000,000, and Poisson's ratio is 0.3. In this region, a downward force is applied, representing the impact force of water flow under actual operating conditions. The analysis focuses on local fatigue, specifically analyzing potential fatigue problems in areas of high stress concentration. (This embodiment focuses on solid mechanics, specifically the blade, not dynamics.)

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

[0099] The coordinate data and stress-strain data are split into three independent data files. The stress-strain data is normalized to construct a regular grid. Normalization helps reduce the impact of data at different scales on model training, ensuring that the contribution of each feature to the model is similar. This embodiment uses a 64×64 grid as an example, interpolating the strain and stress data into the regular grid and converting it into input-output tensors. The cubic method is used for interpolation to smooth the data and better fit any possible nonlinear distributions. Filling null values ​​with fill_value=0 is used to avoid interference from missing values ​​after interpolation on model training. The hyperparameters in this embodiment are set as batchsize=1, modes1=modes2=16, width=64, epochs=1000, lr=0.001. Due to the small amount of data, mini-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: Evaluation of the Geo-FNO prediction model:

[0101] The model was trained using the Adam optimizer and the MSE loss function. The Adam optimizer combines momentum and adaptive adjustment, performing particularly well when handling nonlinear relationships, and exhibits fast and stable training speed. The MSE loss function effectively measures the model's prediction error, making it especially suitable for handling continuous value regression problems, such as predicting physical quantities like stress and strain. Finally, heatmaps for each sample were visualized and saved. The prediction results are shown in the figure below. Figure 1 As shown in the figure, the actual result is as follows. Figure 2 As shown in the figure, the difference graph is as follows: Figure 3 As shown.

[0102] Figure 1 and Figure 2 In the diagram, the X and Y axes represent the spatial coordinates of the object, with the X-axis being the horizontal coordinate and the Y-axis the vertical coordinate. This represents the specific location of the turbine component on a two-dimensional plane. The color bars on the right represent the intensity of the stress value, ranging from 0 to 1. The closer the color is to yellow, the stronger the stress; the closer it is to purple, the weaker the stress. The color gradient in the image shows that the stress is higher in the upper region and lower in the lower region. This indicates that in the upper part of the image, specifically the square area cut by the blades, the upper half is a stress concentration area, corresponding to this area of ​​the component experiencing high stress intensity.

[0103] Figure 3In the graph, the X and Y axes represent the spatial coordinates of the data; the color bars indicate the range of values ​​without corresponding colors, ranging from purple (representing negative values) to yellow (representing positive values). The difference between the predicted and actual values ​​can change from negative to positive, indicating the accuracy of the prediction. A predominance of purple in the lower right area of ​​the graph suggests that the predicted value is lower than the actual value, meaning the model underestimated the result, and its performance needs improvement. Most of the area is green, indicating accurate prediction. The yellow area indicates that the predicted result is higher than the actual value, meaning the model overestimated the result.

Claims

1. A method for interactive virtual-real development of elastic digital twins for turbine components, combining geometric features and Fourier neural operators, characterized in that... The method includes the following steps: Step S1: Setting material properties and boundary conditions: The turbine parts were modeled using finite element analysis software, and key design parameters, including geometric parameters, material property parameters, loads, and boundary conditions, were selected based on their actual working environment and conditions. Step S2, Limited Metadata Generation and Format Conversion: Step S2.1: Generate stress cloud diagrams of the turbine parts before and after deformation through finite element analysis; Step S2.2: Extract the coordinate information of the turbine component nodes and their corresponding stress and strain data, and export the data as .csv format. Then, use a custom script to further convert these data into .npy format. Step S3: Construction and training of the prediction model based on the geometrically perceptual Fourier operator: The data generated in step S2 is preprocessed by geometric feature extraction and Fourier transform to construct a Geo-FNO prediction model, which is then trained and optimized to enable the model to accurately predict the stress distribution and elastic deformation of turbine parts under complex load conditions. Step S4, Model Prediction and Validation: The test data is input into the trained Geo-FNO prediction model to quickly predict the stress distribution and elastic deformation of turbine parts. Then, the prediction accuracy and reliability of the Geo-FNO prediction model are verified by comparing the stress values ​​and deformation amounts in the finite element analysis calculation results.

2. The method for elastic digital twin interaction of turbine parts combining geometric features and Fourier neural operators as described in claim 1, characterized in that... In step S2, the coordinate information of the turbine component nodes and the extraction of their corresponding stress and strain data are completed through the field output interface. The generated data is adapted to the input format requirements of the Geo-FNO model. The specific steps are as follows: (1) Use the post-processing module of the finite element analysis software to generate stress tensor and strain tensor data at the nodes; (2) Based on the topology of the finite element model, extract the global coordinates of the nodes and map them to the mesh cells to ensure the consistency and accuracy of all data points in the computational domain; (3) The extracted data is exported in .csv format by using a custom script and then further converted to .npy format to adapt it to the input requirements of the Geo-FNO model.

3. The method for elastic digital twin interaction of turbine parts combining geometric features and Fourier neural operators as described in claim 1, 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, data standardization is performed to eliminate the influence of physical quantities at different scales on model training and ensure that the data are on the same order of magnitude; then, Fast Fourier Transform (FFT) is used to reduce the dimensionality of the node data of the turbine parts, and the main frequency components of the node data are extracted through FFT. Step S3.1.2, Geometric feature extraction and supplementation: Geometric features of turbine parts are extracted through geometric coding technology and transformed into feature vectors in high-dimensional space. These feature vectors not only contain the shape and size of turbine parts, but also supplement geometric information that may be missed by Fourier transform through geometric transformation. Step S3.2, Geometric and Computational Space Mapping: By transforming coordinates, 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 components under complex load conditions is described by numerically solving the equations. The numerical solution equations are as follows: In the formula, K is the stiffness matrix, containing parameters related to the geometry and material properties of the turbine components, and v represents the vector field of displacement and deformation of the nodes. t Represents the stress field, x represents the spatial coordinate, Fv t It is the external load applied to the model, where F represents the load intensity. This represents the scaling factor, and D represents the physical space; Step S3.4, Frequency Domain Reconstruction: By using inverse Fourier transform, the frequency domain data is converted into a function in physical space, and the stress and deformation physical quantities of the turbine parts are recovered. Step S3.5, Model Training: Train a Geo-FNO-based prediction model to accurately predict the stress distribution and elastic deformation of turbine components under complex load conditions.

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

5. The method for elastic digital twin interaction of turbine parts combining geometric features and Fourier neural operators as described in claim 1, characterized in that... In step S3.1.2, the geometric feature mapping is as follows: X={x1,x2,x3,...,x n } In the formula, x1, x2, x3, ..., x n This indicates the key geometric features of the turbine components.

6. The method for elastic digital twin interaction of turbine parts combining geometric features and Fourier neural operators according to claim 1, characterized in that... In step S3.2, by changing φ a physical space D a The points are mapped to D c Points in computational space: In the formula, D c =[0,1] d It is the standardized computational space.

7. The method for elastic digital twin interaction of turbine parts combining geometric features and Fourier neural operators as described in claim 1, characterized in that... In step S4, the Adam optimizer and mean squared error are used to evaluate the model's prediction performance, and the model is optimized accordingly.

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