Mirror flexible support stress field information reconstruction method based on multitask gaussian process

Abstract

The present application belongs to the field of structure simulation calculation, and particularly relates to a mirror flexible support stress field information reconstruction method of multi-task Gaussian process. Firstly, for a specific mirror flexible support structure, its boundary conditions and load working conditions are determined. The finite element analysis software is used to divide the structure into grids, and a finite element model containing node and element information is generated. Then, a plurality of state input samples are generated in the value range of the load through methods such as Latin hypercube sampling. Part of the nodes in the finite element model is uniformly selected as task input. The training data is obtained through finite element analysis, and a multi-task Gaussian process model based on a graph is constructed. The correlation matrix of the state and the task is calculated, and the log-likelihood function is determined. The hyperparameters of the model are adjusted through an optimization algorithm. For unknown states or unknown tasks that need to reconstruct stress field information, the trained model is used for prediction. According to the prediction result, the stress field information is reconstructed.

Description

Technical Field

[0001] The present invention belongs to the field of structural simulation calculation, and in particular relates to a method for reconstructing stress field information of a flexible support of a reflector using a multi-task Gaussian process. Background Art

[0002] As core components of a space telescope, the performance of the space mirror and its support system directly determines the telescope's key technical specifications. On the one hand, the support system must possess sufficient rigidity to withstand mirror displacement and deformation caused by its own weight and external loads; on the other hand, it must maintain appropriate flexibility to accommodate thermal expansion caused by temperature fluctuations. Flexible support structures are crucial components for maintaining these characteristics. The design of flexible structures requires attention to key performance indicators such as stress and deformation, which rely on the calculation of the structure's full-field performance parameters. Finite element simulation methods can calculate full-field information for a structure under specific operating conditions. However, finite element analysis consumes significant computational time and resources, making it unsuitable for large-scale design solutions. While surrogate models and deep learning methods have applications in physical field reconstruction, their simple structures make it difficult to capture complex variable relationships, resulting in poor performance in complex structural reconstruction. Deep learning methods, on the other hand, have extremely high training data requirements. In the field of flexible mirror support structures, experimental or simulated data often cannot meet these training requirements. Furthermore, their complex network structures consume significant computational resources and lack the ability to predict unknown internal nodes. Summary of the Invention

[0003] The present invention aims to provide a multi-task Gaussian process method for reconstructing the stress field information of the flexible support of a reflector, so as to solve the problems of high computational cost and insufficient reconstruction accuracy in the prior art, and to achieve efficient and accurate reconstruction of the stress field of the flexible support structure of the reflector.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is:

[0005] A multi-task Gaussian process method for reconstructing stress field information of a flexible support of a reflector, comprising the following steps:

[0006] Step 1: Based on the flexible support structure of the reflector, determine the load state variables of the structure. After performing finite element simulation on the flexible structure of the reflector, determine the finite element nodes as the task variables of the multi-task Gaussian process.

[0007] Step 2: Sample the load state variables and nodes using the Latin hypercube sampling method to construct a graph-based multi-task Gaussian process model:

[0008] z(x,y)=f(x) T β+GP(0,C(x,x ′ ,y,y′ ))(1)

[0009] Among them, x and x ′ For the p-dimensional load state input, y and y ′ is the node task input, f(x) T β is the linear regression trend term about the load state input, f(x) represents the function of the load state variable x, and β represents the regression coefficient; GP(0,C(x,x ′ ,y,y ′ )) has a mean of zero and a variance of σ 2 Gaussian process, where C(x,x ′ ,y,y ′ ) is the multi-input correlation coefficient. z(x,y) represents the stress field information of the flexible support structure of the reflector that needs to be reconstructed by the multi-task Gaussian process.

[0010] Step 3: Define the load state and node related functions.

[0011] The load state related function is:

[0012]

[0013] Among them, d s (x,x ′ ) is the distance measure between load states, calculated using the 2-norm. is the hyperparameter of the state-related function.

[0014] The node related functions are:

[0015]

[0016] Among them, the distance d t (y,y ′ ) adopts a graph-based distance metric, which is determined by representing the finite element mesh as an undirected graph and calculating the shortest path between nodes using algorithms such as breadth-first search, so as to better reflect the connectivity and correlation of nodes in the structure. is the hyperparameter of the task-related function.

[0017] Step 4: Define the load state training set X and the node training set Y. Obtain the load state and node correlation matrices R(X,X) and K(Y,Y) through the correlation functions of the load state and the node. The correlation matrix of the graph-based multi-task Gaussian process model is expressed as:

[0018]

[0019] in Represents the Kronecker product of matrices.

[0020] Step 5: Optimize the model's hyperparameters through maximum likelihood estimation.

[0021] The log-likelihood function of the calculated model is:

[0022]

[0023] Where N and M are the number of samples of load state variables and node variables, respectively, z is the training data of stress field information, and F is the vector containing all f(x). 2 and the hyperparameter θ s and θ t Take the derivative and set it to zero to get the corresponding estimated value, thus completing the model training.

[0024] Step 6: Use the trained graph-based multi-task Gaussian process model to reconstruct stress field information under different conditions.

[0025] For the reconstruction of unknown states under known tasks, we define the relevant vector Calculate the log-likelihood function and take its derivative to obtain the predicted value of stress field information for:

[0026]

[0027] For the prediction of known load states at unknown nodes, define the relevant vector Similarly, calculate the log-likelihood function and take its derivative to obtain the predicted value of stress field information. for:

[0028]

[0029] For the prediction of unknown states and unknown tasks, define the relevant vector Calculate the log-likelihood function and take its derivative to obtain the predicted value of stress field information for:

[0030]

[0031] Finally, the stress field information of the flexible support of the reflector can be quickly reconstructed based on the prediction results of these stress field information.

[0032] The effective gain effect of the present invention is:

[0033] A graph-based multi-task Gaussian process model is used to perform full-physics reconstruction of the stress field information of the reflector's flexible support. Compared with traditional multi-task Gaussian process models, this model reduces computational cost and improves efficiency, enabling rapid reconstruction of the stress field of the reflector's flexible support structure. Introducing a graph-based distance metric to construct task relevance better adapts to the complex spatial connectivity of the reflector's flexible support structure, accurately capturing the correlations between nodes within the structure, thereby improving the accuracy of stress field reconstruction and effectively addressing the stress field reconstruction problem in complex structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 This is a flow chart of the method for reconstructing stress field information of a flexible support of a reflector using a multi-task Gaussian process of the present invention.

[0035] Figure 2 This is a schematic diagram of modeling the flexible support structure of the reflector in an embodiment of the present invention; in the figure, 1 is a side support, 2 is a reflector, 3 is a mirror chamber, and 4 is a flexible support.

[0036] Figure 3 This is the boundary condition of the flexible support structure of the reflector in an embodiment of the present invention.

[0037] FIG4(a) and FIG4(b) are mesh division and finite element result diagrams of the flexible support structure of the reflector according to an embodiment of the present invention.

[0038] Figure 5 This is the Pareto front result of the embodiment of the present invention. DETAILED DESCRIPTION

[0039] The present invention will be further described below with reference to specific embodiments.

[0040] like Figure 1 As shown, the present invention provides a method for reconstructing the stress field information of a flexible support of a reflector based on a multi-task Gaussian process model of a graph. First, for a specific flexible support structure of a reflector, its boundary conditions and load conditions are determined. Finite element analysis software is used to mesh the structure to generate a finite element model containing node and unit information. Then, multiple state input samples are generated within the range of load values ​​through methods such as Latin hypercube sampling. Some nodes are uniformly selected from the nodes of the finite element model as task inputs. Training data is obtained through finite element analysis. A multi-task Gaussian process model based on a graph is constructed based on the training data obtained above. The correlation matrix of the state and task is calculated, and the log-likelihood function is determined. The hyperparameters of the model are adjusted by an optimization algorithm to maximize the log-likelihood function to complete the training of the model. For unknown states or unknown tasks that require stress field reconstruction, the trained model is used for prediction. Based on the prediction results, a stress field distribution cloud map of the flexible support structure of the reflector is drawn to achieve stress field reconstruction.

[0041] Figure 2 Schematic diagram of the modeling of the flexible support structure of the reflector in the embodiment. 1 is the side support of the reflector, 2 is the reflector, 3 is the mirror chamber where the reflector support structure is placed, and 4 is the flexible support structure of this embodiment. The boundary conditions of the flexible support structure of the reflector are as follows: Figure 3 As shown in Figure 4, the lower end of the flexible support is fixedly constrained, and the upper surface is subject to a force controlled in three directions. The force ranges are [-100, 100], [-200, -50], and [-100, 100], respectively. Figures 4(a) and 4(b) show the meshing results of the flexible support structure and the stress results from the finite element analysis under the action of a certain force. The mesh consists of 50,982 elements and 78,211 nodes.

[0042] The process of the method for reconstructing the stress field information of the flexible support of the reflector according to the embodiment of the present invention is as follows:

[0043] First, the Latin hypercube sampling method was used to sample the design variables of the three forces, generating 20 state input samples. The number of state training samples was set to 15, and the number of state test samples was set to 5. Task nodes were evenly selected on the surface of the flexible support structure as task training data, resulting in a total of 473 nodes. The remaining nodes were used as test data. Finite element analysis software was used to accurately calculate the stress response of the entire structure for each state input sample, thereby obtaining high-quality training and test data.

[0044] Afterwards, a graph-based multi-task Gaussian process model is constructed based on the collected training data.

[0045] Finally, the stress field is reconstructed using a graph-based multi-task Gaussian process. Three experiments are set up, namely Experiment 1: predicting the 5 unknown states at 473 nodes, Experiment 2: predicting the 15 training states of the remaining nodes outside the 473 nodes, and Experiment 3: predicting the 5 unknown states of the remaining nodes outside the 473 nodes. 2 The reconstruction results are displayed, such as Figure 5 As shown, we can see that the R 2 The prediction results are good, R 2 The values ​​all exceed 0.5. Experiment 2 yields the best reconstruction results, followed by Experiment 1, and Experiment 3 has the worst. This is because Experiment 3 predicts unknown states in an unknown task, which contains the most unknown information compared to the other two experiments. Therefore, the prediction accuracy is relatively poor.

Claims

1. A multi-task Gaussian process method for reconstructing stress field information of flexible supports of reflectors, characterized in that: Here are the steps: Step 1: Based on the flexible support structure of the reflector, determine the load state variables of the structure. After performing finite element simulation on the flexible structure of the reflector, determine the finite element nodes as the task variables of the multi-task Gaussian process. Step 2: Sample the load state variables and nodes using the Latin hypercube sampling method to construct a graph-based multi-task Gaussian process model: z(x,y)=f(x) T β+GP(0,C(x,x ′ ,and,and ′ ))(1) Among them, x and x ′ For the p-dimensional load state input, y and y ′ is the node task input, f(x) T β is the linear regression trend term about the load state input, f(x) represents the function of the load state variable x, and β represents the regression coefficient; GP(0,C(x,x ′ ,y,y ′ )) has a mean of zero and a variance of σ 2 Gaussian process, where C(x,x ′ ,y,y ′ ) is the multi-input correlation coefficient; z(x,y) represents the stress field information of the flexible support structure of the reflector that needs to be reconstructed by the multi-task Gaussian process; Step 3: Define the relevant functions of load state and node; The load state related function is: Among them, d s (x,x ′ ) is the distance measure between load states, calculated using the 2-norm; is the hyperparameter of the state-related function; The node related functions are: Among them, the distance d t (y,y ′ ) adopts a graph-based distance metric, which is determined by representing the finite element mesh as an undirected graph and calculating the shortest path between nodes using algorithms such as breadth-first search; is the hyperparameter of the task-related function; Step 4: Define the load state training set X and the node training set Y. Obtain the load state and node correlation matrices R(X,X) and K(Y,Y) through the correlation functions of the load state and the node. The correlation matrix of the graph-based multi-task Gaussian process model is expressed as: in represents the Kronecker product of matrices; Step 5: Optimize the model's hyperparameters through maximum likelihood estimation; The log-likelihood function of the calculated model is: Among them, N and M are the number of samples of load state variables and node variables, z is the training data of stress field information, and F is the vector containing all f(x); the log-likelihood function is transformed into the regression coefficient β and variance σ. 2 and the hyperparameter θ s and θ t Take the derivative and set it to zero to get the corresponding estimated value, thus completing the model training; Step 6: Use the trained graph-based multi-task Gaussian process model to reconstruct stress field information under different conditions; For the reconstruction of unknown states under known tasks, we define the relevant vector Calculate the log-likelihood function and take its derivative to obtain the predicted value of stress field information for: For the prediction of known load states at unknown nodes, define the relevant vector Similarly, calculate the log-likelihood function and take its derivative to obtain the predicted value of stress field information. for: For the prediction of unknown states and unknown tasks, define the relevant vector Calculate the log-likelihood function and take its derivative to obtain the predicted value of stress field information for: Finally, the stress field information of the flexible support of the reflector can be quickly reconstructed based on the prediction results of these stress field information.

Images