A turbine blade precision casting stress prediction system and method based on digital twinning

Abstract

The present application belongs to the technical field of investment precision casting, and particularly relates to a turbine blade precision casting stress prediction system and method based on digital twinning. The system comprises a client for realizing sensor data acquisition and visual rendering in the turbine blade precision casting process; a server for realizing training on stress field finite element simulation data to obtain a proxy model; the client and the server realize data connection and information interaction through Socket communication to jointly constitute a digital twinning model of the turbine blade; and the digital twinning model is updated and optimized by using the sensor data collected by the client. Stress cloud map visualization is realized by using Unity3D, and the computer real-time rendering overhead is reduced based on the RBF interpolation method to solve the problems of real-time stress measurement difficulty and visualization impossibility in the turbine blade precision casting process. Compared with the average absolute error of 435.42 MPa of the finite element simulation stress prediction, the average absolute error of the present application is 97.05 MPa, which can effectively reduce the error and improve the prediction accuracy.

Description

Technical Field

[0001] This invention belongs to the field of investment casting technology, specifically a stress prediction system and method for turbine blade precision castings based on digital twins. Background Technology

[0002] Turbine blades are high-strength, thin-walled structural components composed of numerous free-form surfaces and complex hollow cavities. They are typically manufactured using investment casting, representing the pinnacle of materials forming and precision casting technology. However, domestic aero-engine applications indicate that turbine blade precision casting still suffers from unstable quality and high scrap rates. Common quality issues include dimensional deviations, shrinkage cavities, and cracks. Directed columnar and single-crystal blades are also prone to casting defects such as impurities and recrystallization. Research indicates that solidification shrinkage, machining, and high-temperature service can induce stress and strain in directed columnar and single-crystal turbine blades. High-density dislocations formed by localized strain reorganize at high temperatures, resulting in stress concentration in the affected area. When the critical stress is exceeded, recrystallization can occur, forming transverse grain boundaries perpendicular to the principal stress, leading to a sharp decline in blade performance. Under thermal load impact, this can even cause premature blade fracture, resulting in serious aviation accidents. Therefore, research on residual stress in turbine blades is of great significance.

[0003] The testing and verification of residual stress in single-crystal and directional columnar crystal turbine blades is extremely complex and inefficient. Obtaining the residual stress distribution of precision-cast blades solely through measurement is even more difficult and time-consuming. Therefore, it is essential to model and predict the overall residual stress distribution of precision-cast blades based on a limited number of measurement points.

[0004] Current research on residual stress in turbine blades primarily relies on finite element simulation (FEM). FEM can save on testing costs, optimize testing processes, and predict test result trends, thus improving the success rate of experiments to some extent. However, complex physical systems are often difficult to model accurately, making it impossible to assess their state and optimize control through analytical mathematical models. For example, existing technologies include methods for predicting the thermal barrier coating stress of turbine blades and predicting residual stress from blade grinding. Although these methods describe turbine blade stress prediction approaches, they are all finite element simulations performed for specific scenarios or functions.

[0005] As the background information above indicates, research on residual stress in turbine blades is of great significance. However, stress measurement of single-crystal and oriented columnar crystal turbine blades is extremely complex and inefficient, typically requiring four hours to measure a single point. Therefore, only a limited number of measurement points can be obtained, which cannot reflect the overall stress distribution of the blade. Current research on residual stress in turbine blades primarily relies on finite element simulation (FEM). While FEM can characterize the overall stress distribution of the blade, simulation results often do not match actual measurements. Improving prediction accuracy can be achieved by resimulating the blade by changing boundary conditions and back-calculating heat transfer coefficients. However, resimulation is extremely time-consuming, taking at least two days per simulation, and even slight modifications require rerunning the simulation, resulting in significant repetitive work and consuming considerable time and computational resources. Furthermore, it is difficult to establish accurate mathematical models for complex physical systems, making it challenging to guarantee the accuracy and reliability of FEM simulations.

[0006] Therefore, it is necessary to provide a stress prediction system and method for turbine blade precision castings based on digital twins in order to solve the above problems. Summary of the Invention

[0007] The technical problem to be solved by this invention:

[0008] To address the challenges of complex and inefficient real-time stress measurement of single-crystal and oriented columnar turbine blades, which fails to capture the overall stress distribution, and the limitations of finite element simulation in ensuring accuracy and reliability, this invention proposes a stress prediction system and method for precision-cast turbine blades based on digital twins. Utilizing finite element simulation data, historical data, and real-time sensor data, the system updates and optimizes the prediction proxy model, making its stress prediction results more closely match the actual state of the blade. This achieves real-time, efficient, and visualized stress measurement, ensuring both accuracy and reliability. Digital twin technology pre-trains the stress prediction proxy model and incorporates a small amount of measured data to correct it online, eliminating the need for retraining each time, reducing repetitive work, saving time and computational resources, and achieving prediction accuracy superior to finite element simulation.

[0009] A stress prediction system for turbine blade precision casting based on digital twins includes a client for acquiring sensor data and visualizing the turbine blade precision casting process; and a server for training finite element simulation data of the stress field to obtain a proxy model for stress prediction in the turbine blade precision casting process.

[0010] The client and server communicate via Socket to establish data connection and information exchange, jointly forming a digital twin model of the turbine blade. The digital twin is updated and optimized using sensor data collected by the client to complete the stress prediction of the precision-cast turbine blade.

[0011] A further technical solution of the present invention is: the server exports stress field data from the post-processing module of the finite element software, performs RBF interpolation, and trains a surrogate model for stress prediction.

[0012] A further technical solution of the present invention is: the client-side visualization rendering is built on Unity3D, the three-dimensional geometric model of the turbine blade is exported into STL format, which contains a three-dimensional entity composed of triangular facets, the XYZ coordinates of each triangular facet are used as input, the equivalent stress value is predicted by the proxy model trained on the server as output, and the stress field is visualized and rendered.

[0013] A further technical solution of the present invention is that the stress field data includes the XYZ coordinates of each grid node and the equivalent stress value.

[0014] A further technical solution of the present invention is: for each grid node, the RBF interpolation algorithm uses the KNN algorithm to find its nearest N neighboring nodes, where N is greater than or equal to 15, and interpolates the stress value of the node, thereby realizing RBF interpolation for all nodes.

[0015] A further technical solution of the present invention is: different colors are used to render the stress field for stress values ​​of different magnitudes.

[0016] A method for stress prediction of precision-cast turbine blades based on digital twins includes the following steps:

[0017] Step 1: Build a stress prediction system for turbine blade precision casting based on digital twin based on C / S architecture. The client realizes sensor data acquisition during the precision casting process of turbine blade, and the server trains the stress field finite element simulation data to obtain a proxy model for stress prediction in the precision casting process of turbine blade.

[0018] Step 2: The client and server establish data connection and information interaction through Socket communication, and realize visualization based on Unity3D. The proxy model trained in Step 1 is encapsulated in the server to establish a digital twin model for stress prediction in the turbine blade precision casting process.

[0019] Step 3: Based on the digital twin model for stress prediction in the precision casting process of turbine blades established in Step 2, update and optimize the digital twin model using sensor data collected from the production site. Once the preset conditions are met, complete the stress prediction of the precision casting of turbine blades.

[0020] A further technical solution of the present invention is: in step 3, the function representation process of the digital twin model is as follows:

[0021] The true stress state of the blade is represented by the function y = f(x), and the data points {x} within the domain are given. i ,yi Strictly satisfy y i =f(x) i ), i = 1,...,N, the above data points are called "control nodes" or "interpolation nodes";

[0022] Digital twin models are based on interpolation functions It means that, among them,

[0023] Radial basis functions are introduced as basis functions for the interpolation function, and are defined as follows:

[0024]

[0025] In the formula, σ represents the standard deviation of the variable;

[0026] The digital twin model can then be represented by the RBF interpolation function:

[0027]

[0028] In the formula, ω i These are the coefficients or weights of each interpolation node;

[0029] Substituting (1) into (2), we get:

[0030]

[0031] in It is an interpolation matrix, and this interpolation matrix is ​​symmetric; for the Gaussian function, the diagonal elements of the interpolation matrix have a value of 1;

[0032] Let the system of linear equations be denoted as ΦW=Y, and the j-th row of this system of equations is:

[0033]

[0034] Therefore, the coefficients of the RBF interpolation function are calculated as W = Φ. -1 Y is then used to obtain the functional representation of the digital twin model.

[0035] A further technical solution of the present invention is: step 1 includes the following method steps:

[0036] Step 1.1: Build a stress prediction system for turbine blade precision casting based on digital twin based on C / S architecture. The client is set up at the actual production site of turbine blades and collects stress data during the precision casting process of turbine blades through sensors.

[0037] Step 1.2: Perform full-element modeling and finite element simulation of the turbine blade precision casting process, including the casting process model, gating system model, ceramic core and cold copper model, and virtual geometric model after assembly to generate the mold shell; use finite element software to simulate the physical model of its temperature field, stress field, and displacement field changes; construct a behavioral model of its multi-domain dynamic evolution of mold shell and ceramic core constraint removal operations; and set the rule model for simulation boundary conditions and process parameters; through full-element modeling and simulation of the turbine blade precision casting process, obtain the finite element simulation data of the stress field during the turbine blade precision casting process.

[0038] Step 1.3: Use the RBF surrogate model to train the stress field finite element simulation data. The X, Y, and Z coordinates of the turbine blade finite element mesh nodes are used as inputs, and the stress values ​​are used as outputs to obtain a surrogate model for stress prediction in the turbine blade precision casting process.

[0039] A further technical solution of the present invention is: step 2 includes the following method steps:

[0040] Step 2.1: The client and server establish data connection and information exchange through Socket communication;

[0041] Step 2.2: Visualize the stress prediction results of turbine blades based on Unity3D, and render stress cloud maps based on finite element mesh nodes;

[0042] Step 2.3: Save the trained proxy model from Step 1 as a PKL file and encapsulate it on the server to establish a digital twin model for stress prediction in the precision casting process of turbine blades.

[0043] A further technical solution of the present invention is: in step 3, the preset condition is: the stress prediction result approximates the actual state of the blade.

[0044] A further technical solution of the present invention is: in step 1.1, the sensor is an X-ray diffractometer or a strain gauge.

[0045] In step 1.2, the finite element software used is ProCAST.

[0046] Beneficial effects

[0047] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0048] 1. This invention provides a stress prediction system and method for turbine blade precision casting based on digital twins. It can perform full-element modeling and simulation of the turbine blade precision casting process, completely simulating the stress changes of the turbine blade from the start of solidification to the end of unconstraint removal. By establishing a digital twin model of the turbine blade precision casting process, and combining finite element simulation data, historical operation data, and sensor measurement data, the prediction proxy model is updated and optimized, making its stress prediction results closer to the actual state of the blade. Compared to the average absolute error of 435.42 MPa in finite element simulation stress prediction, the average absolute error of the digital twin model stress prediction is 97.05 MPa, effectively reducing errors and improving prediction accuracy.

[0049] 2. This invention provides a stress prediction system and method for precision-cast turbine blades based on digital twins. It adopts a C / S architecture (client / server architecture), with the client written in C# using Unity3D and the server written in Python. Stress field data is exported from the post-processing module of finite element simulation software. The data format consists of the XYZ coordinates of mesh nodes and their equivalent stress values, which are used to construct a training set. The 3D geometric model of the turbine blade to be rendered is exported as an STL format (3D solid composed of triangular facets). For each triangular facet vertex, its XYZ coordinates are used as input. The KNN algorithm is used to find the N nearest neighbor nodes in the training set. Based on these N nodes, the RBF interpolation algorithm is used to interpolate the stress value of each mesh node, thus achieving RBF interpolation prediction for all nodes. Different colors are assigned according to the predicted stress values ​​to render a stress cloud map. Unity3D is used for visualization of the stress cloud map. The RBF interpolation method reduces the real-time rendering overhead of the computer, solving the problems of difficult real-time stress measurement and visualization during the precision casting process of turbine blades. Attached Figure Description

[0050] Figure 1 This is a block diagram of the digital twin system of the present invention;

[0051] Figure 2 The specific implementation process of full-element modeling and finite element simulation for the precision casting process of turbine blades;

[0052] Figure 3 This is a visualization rendering of the stress cloud diagram of the turbine blade digital twin of the present invention;

[0053] Figure 4 The location distribution of the stress measurement points in this invention;

[0054] Figure 5 This is a comparison chart of the stress prediction results of the digital twin model of the present invention. Detailed Implementation

[0055] To more clearly illustrate the technical solutions implemented in this invention, the various modules required in the embodiments will be briefly described below. Obviously, the accompanying drawings described below are merely flowcharts of this invention. Those skilled in the art can extend the scope of the invention based on these drawings without any creative effort. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0056] Example 1

[0057] like Figure 1 The digital twin system shown includes a server that is used to train finite element simulation data of stress field to obtain a surrogate model for stress prediction in the precision casting process of turbine blades.

[0058] Digital twin technology uses simulation data, historical data, and real-time sensor data to update, correct, connect, and supplement mathematical models, enabling the model's predictions to better match the stress distribution during the blade precision casting process and to better evaluate the system in real time.

[0059] Specifically, stress field data is exported from the post-processing module of the finite element software ProCAST, including the XYZ coordinates and equivalent stress values ​​(in MPa) of each grid node. For each grid node, the KNN algorithm is used to find its 15 nearest neighboring nodes, and the stress value of this node is interpolated using the RBF interpolation algorithm. RBF interpolation is performed on all nodes to train a surrogate model for stress prediction.

[0060] The client is used to acquire sensor data and visualize the turbine blades during the precision casting process. Specifically, it uses C# and Unity3D to build the client, which exports the three-dimensional geometric model of the turbine blades into STL format, i.e., a three-dimensional solid composed of triangular facets. The XYZ coordinates of each triangular facet are used as input, and the equivalent stress value is predicted by the proxy model trained on the server as the output. Different colors are used to render the stress field for different stress values.

[0061] The client and server communicate via Socket to connect and exchange data, forming a digital twin of the turbine blade. After the client collects measured stress data from the X-ray diffractometer, it transmits the coordinates and stress value of the point to the server via Socket communication. The server preprocesses the data into the same format as the training set, and then uses the KNN algorithm to replace the stress values ​​of the 15 points closest to the measured value, completing the data fusion. Through this virtual-real interaction, the proxy model is updated, making the stress prediction results of the digital twin more closely resemble the actual state of the blade.

[0062] A method for stress prediction of precision-cast turbine blades based on digital twins, the implementation steps of which are as follows:

[0063] Step 1: Client ① collects sensor data during the precision casting process of turbine blades, and server ② trains the stress field finite element simulation data to obtain a proxy model for stress prediction during the precision casting process of turbine blades.

[0064] Step 1.1: Build a digital twin system for stress prediction in the precision casting process of turbine blades based on C / S architecture. The client ① is set at the actual production site of turbine blades and collects stress data in the precision casting process of turbine blades through sensors such as X-ray diffractometers and strain gauges.

[0065] Step 1.2: Perform full-element modeling and finite element simulation of the turbine blade precision casting process③, including the casting process model, gating system model, ceramic core and cold copper model, and virtual geometric model after assembly to generate the mold shell④; use finite element software such as ProCAST to simulate the physical model of its temperature field, stress field, and displacement field changes⑤; construct a behavioral model of its dynamic evolution in multiple fields such as mold shell removal and ceramic core removal constraint operations⑥; and set the rule model for simulation boundary conditions and process parameters⑦; the specific implementation process of full-element modeling and finite element simulation is as follows: Figure 2 As shown, by performing full-element modeling and simulation of the turbine blade precision casting process, finite element simulation data of the stress field during the turbine blade precision casting process are obtained.

[0066] Step 1.3: Use the RBF surrogate model to train the stress field finite element simulation data. The X, Y, and Z coordinates of the turbine blade finite element mesh nodes are used as inputs, and the stress values ​​are used as outputs to obtain the surrogate model for stress prediction in the turbine blade precision casting process.

[0067] Step 2: The client and server establish data connection and information exchange through Socket communication⑨, and realize visualization based on Unity3D to build a digital twin model⑩ for stress prediction in the precision casting process of turbine blades;

[0068] Step 2.1: The client and server establish data connection and information exchange through Socket communication (9).

[0069] Step 2.2: Visualize the turbine blade stress prediction results using Unity3D. Render stress cloud maps based on the finite element mesh nodes, such as... Figure 4 As shown;

[0070] Step 2.3: Save the trained proxy model from Step 1 as a PKL file and encapsulate it on the server to establish a digital twin model for stress prediction in the precision casting process of turbine blades.

[0071] Assuming the true stress state of the blade can be represented by the function y = f(x), some data points {x} within the domain... i ,y i Strictly satisfy y i =f(x) i Let i = 1, ..., N. These data points are called "control nodes" or "interpolation nodes". If there exists a function with a relatively simple form (such as an nth-degree polynomial)... Make If all of these conditions are met, then a digital twin model that approximates the actual stress state of this blade can be derived from the interpolation function. express.

[0072] Radial basis functions are introduced as basis functions for the interpolation function, and are defined as follows:

[0073]

[0074] In equation (1), σ represents the standard deviation of the variable, which describes the degree of dispersion among individual variables;

[0075] The digital twin model can then be represented by the RBF interpolation function:

[0076]

[0077] In the formula, ω i These are the coefficients or weights of each interpolation node;

[0078] Substituting (1) into (2), we get:

[0079]

[0080] in It is an interpolation matrix, therefore the interpolation matrix is ​​symmetric. For the Gaussian function, the diagonal elements of the interpolation matrix have a value of 1.

[0081] Let the system of linear equations be denoted as ΦW=Y, and the j-th row of this system of equations is:

[0082]

[0083] Therefore, the coefficients of the RBF interpolation function can be calculated as W = Φ. -1 Y can then be used to obtain the functional representation of the digital twin model.

[0084] Step 3: Based on the digital twin model for stress prediction in the precision casting process of turbine blades established in Step 2, update and optimize the digital twin model using sensor data collected from the production site, so that the stress prediction results continuously approach the actual state of the blades.

[0085] Example 2

[0086] The effects of this invention can be further illustrated by the following experiments.

[0087] 1. Experimental conditions

[0088] The residual stress in the turbine blade casting was measured using a Proto MG2000 X-ray diffractometer. Alternatively, conventional stress measurement equipment such as strain gauges and neutron diffractometers can also be used. The precision casting process of the turbine blade was simulated using the finite element simulation software ProCAST.

[0089] 2. Experimental Data

[0090]

[0091] 3. Experiment Content

[0092] A digital twin of the turbine blade was constructed using the method described above. Nine measuring points were selected to measure stress values. The locations of the measuring points are shown in the attached figure. Figure 3 As shown in the image. A visualization rendering of the turbine blade stress prediction results, as shown in the image. Figure 4 As shown.

[0093] Analysis of experimental results: such as Figure 5 As shown, the maximum absolute error between the finite element simulation stress prediction results and the measured results is 816.77 MPa, the minimum absolute error is 33.13 MPa, and the average absolute error is 435.42 MPa; the maximum absolute error between the digital twin model stress prediction results and the measured results is 221.90 MPa, the minimum absolute error is 0.439 MPa, and the average absolute error is 97.05 MPa. Therefore, the method described in this invention can significantly improve the prediction accuracy.

[0094] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A stress prediction system for precision-cast turbine blades based on digital twins, characterized in that: It includes a client-side application for acquiring sensor data and visualizing it during the precision casting process of turbine blades; and a server-side application for training finite element simulation data of the stress field to obtain a proxy model for stress prediction during the precision casting process of turbine blades. A stress prediction system for turbine blade precision castings based on digital twins was built using a client / server architecture. The client is located at the actual turbine blade production site, and stress data during the precision casting process is collected using an X-ray diffractometer or strain gauge. The system performs full-element modeling and finite element simulation of the turbine blade precision casting process, including casting process models, gating system models, ceramic core and cold copper models, and virtual geometric models after assembly to generate the mold shell. A physical model simulating the changes in temperature, stress, and displacement fields is used with finite element software. A behavioral model of the dynamic evolution of multi-domain processes during mold shell and ceramic core removal constraints is constructed. A rule model for setting simulation boundary conditions and process parameters is also included. Through full-element modeling and simulation of the turbine blade precision casting process, finite element simulation data of the stress field during the precision casting process is obtained. The RBF surrogate model was used to train the stress field finite element simulation data. The X, Y, and Z coordinates of the turbine blade finite element mesh nodes were used as inputs and the stress values ​​were used as outputs to obtain a surrogate model for stress prediction in the precision casting process of turbine blades. The client and server communicate via Socket to establish data connection and information exchange, jointly forming a digital twin model of the turbine blade; the digital twin is updated and optimized using sensor data collected by the client to complete the stress prediction of the precision-cast turbine blade. The functional representation of the digital twin model is as follows: The true stress state of the blade is expressed by a function. This indicates the data points within the defined domain. Strictly meet The aforementioned data points are called "control nodes" or "interpolation nodes"; Digital twin models are based on interpolation functions It means that, among them, ; Radial basis functions are introduced as basis functions for the interpolation function, and are defined as follows: (1) In formula (1) The standard deviation represents the variance of a variable and describes the degree of dispersion among individual variables. The digital twin model can then be represented by the RBF interpolation function: (2) In the formula, These are the coefficients or weights of each interpolation node; Substituting (1) into (2), we get: (3) in , It is an interpolation matrix; Let the system of linear equations be denoted as The j-th line of this system of equations is: (4) Therefore, the coefficients of the RBF interpolation function are obtained as follows: Furthermore, the functional representation of the digital twin model is obtained.

2. The stress prediction system according to claim 1, characterized in that: The server exports stress field data from the finite element software post-processing module, performs RBF interpolation, and trains a surrogate model for stress prediction.

3. The stress prediction system according to claim 1, characterized in that: The client-side visualization rendering is built on Unity3D. The three-dimensional geometric model of the turbine blade is exported into STL format, which contains a three-dimensional solid composed of triangular facets. The XYZ coordinates of each triangular facet are used as input, and the equivalent stress value is predicted by the proxy model trained on the server as output, and the stress field is visualized and rendered.

4. The stress prediction system according to claim 1, characterized in that: The stress field data includes the XYZ coordinates of each grid node and the equivalent stress value.

5. The stress prediction system according to claim 1, characterized in that: The RBF interpolation algorithm uses the KNN algorithm to find the N nearest neighboring nodes for each grid node, where N is greater than or equal to 15, and interpolates the stress value of that node, thereby achieving RBF interpolation for all nodes.

6. The stress prediction system according to claim 1, characterized in that: Different stress values ​​are rendered using different colors to represent the stress field.

7. A method for stress prediction of precision-cast turbine blades based on digital twins, characterized in that: Includes the following steps: Step 1: Build a stress prediction system for turbine blade precision casting based on digital twin based on C / S architecture. The client realizes sensor data acquisition during the precision casting process of turbine blade, and the server trains the stress field finite element simulation data to obtain a proxy model for stress prediction in the precision casting process of turbine blade. Step 1.1: Build a stress prediction system for turbine blade precision casting based on digital twin based on C / S architecture. The client is set at the actual production site of turbine blades and stress data during the precision casting process of turbine blades are collected by X-ray diffractometer or strain gauge. Step 1.2: Perform full-element modeling and finite element simulation of the turbine blade precision casting process, including the casting process model, gating system model, ceramic core and cold copper model, and virtual geometric model after assembly to generate the mold shell; use finite element software to simulate the physical model of its temperature field, stress field, and displacement field changes; construct a behavioral model of its multi-domain dynamic evolution of mold shell and ceramic core constraint removal operations; and set the rule model for simulation boundary conditions and process parameters; through full-element modeling and simulation of the turbine blade precision casting process, obtain the finite element simulation data of the stress field during the turbine blade precision casting process. Step 1.3: Use the RBF surrogate model to train the stress field finite element simulation data. The X, Y, and Z coordinates of the turbine blade finite element mesh nodes are used as inputs, and the stress values ​​are used as outputs to obtain a surrogate model for stress prediction in the turbine blade precision casting process. Step 2: The client and server establish data connection and information interaction through Socket communication, and realize visualization based on Unity3D. The proxy model trained in Step 1 is encapsulated in the server to establish a digital twin model for stress prediction in the turbine blade precision casting process. Step 3: Based on the digital twin model for stress prediction in the precision casting process of turbine blades established in Step 2, update and optimize the digital twin model using sensor data collected at the production site. After reaching the preset conditions, complete the stress prediction of the precision casting of turbine blades. The functional representation of the digital twin model is as follows: The true stress state of the blade is expressed by a function. This indicates the data points within the defined domain. Strictly meet The aforementioned data points are called "control nodes" or "interpolation nodes"; Digital twin models are based on interpolation functions It means that, among them, ; Radial basis functions are introduced as basis functions for the interpolation function, and are defined as follows: (1) In formula (1) The standard deviation represents the variance of a variable and describes the degree of dispersion among individual variables. The digital twin model can then be represented by the RBF interpolation function: (2) In the formula, These are the coefficients or weights of each interpolation node; Substituting (1) into (2), we get: (3) in , It is an interpolation matrix; Let the system of linear equations be denoted as The j-th line of this system of equations is: (4) Therefore, the coefficients of the RBF interpolation function are obtained as follows: Furthermore, the functional representation of the digital twin model is obtained.

8. The stress prediction method according to claim 7, characterized in that: Step 2 includes the following method steps: Step 2.1: The client and server establish data connection and information exchange through Socket communication; Step 2.2: Visualize the stress prediction results of turbine blades using Unity3D, and render stress cloud maps based on finite element mesh nodes; Step 2.3: Save the trained proxy model from Step 1 as a PKL file and encapsulate it on the server to establish a digital twin model for stress prediction in the precision casting process of turbine blades.

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