Intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessel

Abstract

The invention belongs to the technical field of pressure vessel structure analysis, and particularly relates to an intelligent reduced basis finite element method for rapidly analyzing the deformation behavior of a pressure vessel. According to the method, a finite element method is improved by adopting a reduced basis method, an online and offline combined solution mode is developed, after design parameters such as a structure size, a load type and size, material attributes and the like are preset, a reduced basis space is obtained by adopting an offline solution mode, and then a mapping relation between a reduced basis solution and a finite element solution is established; according to the method, order reduction of a finite element model is achieved, the degree of freedom of a problem is reduced, the calculation amount needed by solving a linear algebraic equation set through a traditional finite element method is greatly reduced while the precision is guaranteed, a Gaussian process regression method is further introduced to describe the elastic-plastic deformation behavior of the pressure vessel, and therefore the deformation behavior of the pressure vessel is rapidly solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of pressure vessel structure analysis, and in particular is an intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels. Background Art

[0002] Pressure vessels refer to container equipment that can withstand a certain internal pressure load and are widely used in various important industrial fields such as petrochemicals, nuclear industry, aerospace, etc. Since pressure vessels often contain flammable and explosive dangerous goods, and are subjected to multiple loads such as pressure, deadweight, and pipe load during operation, failure behavior will cause great danger. Therefore, studying its stress state and deformation behavior under load is of great significance for the design, safety assessment, operation and maintenance of pressure vessels. At present, the finite element method, as a very effective numerical method, plays an important role in the safety assessment and mechanical behavior analysis of pressure vessels.

[0003] Finite element analysis (FEM) is often used in the structural design and safety analysis of pressure vessels. During structural design optimization, it is necessary to continuously vary the design parameters of the pressure vessel and perform simulations to identify the optimal design solution based on specific objectives. Furthermore, during the operation of large pressure vessels, FEM can also be used to simulate and analyze the deformation behavior and stress state of the pressure vessel under certain loads. During pressure vessel optimization and operational simulation, repeated FEM analysis of the structure with multiple variations of relevant design variables or load parameters is necessary, consuming significant computational resources and time. The computational speed of FEM often depends on the number of meshes and nodes. Analyzing pressure vessel models with large meshes requires significant computational resources and time. Furthermore, pressure vessels may exhibit nonlinear elastic-plastic deformation behavior during long-term service or under heavy loads, requiring iterative solutions for numerical analysis, further increasing the computational complexity. Consequently, numerical methods such as FEM still have significant limitations in the design and analysis of complex pressure vessels, especially for real-time monitoring of deformation behavior and stress states. Therefore, it is necessary to develop an efficient numerical analysis method to reduce the simulation calculation time required for each change in structure or load parameters, so as to achieve rapid simulation analysis of the mechanical behavior of pressure vessels.

[0004] Improving the computational efficiency of finite element algorithms can be considered from two perspectives: improving computer performance and reducing computational degrees of freedom. Improving computer performance is costly and overly dependent on advances in computer hardware technology. The conventional method of reducing degrees of freedom by increasing the mesh size reduces computational accuracy and, consequently, the reliability of the results. The reduced basis method (RBM) uses the Galerkin method to map the high-dimensional solution space to a low-dimensional space, significantly reducing the dimensionality and computational complexity of the linear algebraic equations. Reliable spatial mapping error estimation ensures the accuracy of the results, enabling accurate and rapid solution of the equations. RBM has great potential for problems where the governing equations remain constant while the physical parameters, solution domain, and boundary conditions are constantly changing. However, conventional RBMs connect the solution space and the low-dimensional space through linear mappings, making them difficult to apply to nonlinear problems such as elastic-plastic deformation, common in pressure vessels. Summary of the Invention

[0005] To avoid and overcome the technical problems existing in the prior art, the present invention provides an intelligent reduced-basis finite element method for rapid analysis of pressure vessel deformation behavior. The present invention effectively reduces computational complexity, thereby effectively improving the efficiency of generating finite element simulation results.

[0006] To achieve the above object, the present invention provides the following technical solutions: An intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels comprises the following steps: Step 1: Establish a pressure vessel geometry model for finite element simulation, and use the pressure vessel geometry model to calculate the finite element simulation results corresponding to each set of design parameters; Step 2: Construct a reduced basis space based on the finite element simulation results, perform Schmidt orthogonalization on the reduced basis space to obtain orthogonal space basis vectors; calculate the reduced basis solution through the orthogonal space basis vectors, construct the reduced basis solution space, and construct the design parameter space based on the training samples; Step 3: Use the Gaussian process regression method to establish a mapping relationship between the design parameter space and the reduced basis solution space. At the same time, select the square exponential kernel function determined by auto-correlation as the kernel function of the Gaussian regression process. Use maximum likelihood estimation to optimize the kernel function through parameter optimization. Based on the optimized kernel function, establish a Gaussian process regression model that characterizes the nonlinear mapping relationship between the reduced basis solution and the design parameters. Step 4: Train the Gaussian process regression model using the finite element simulation results until the prediction accuracy requirements are met; and use the trained Gaussian process regression model to predict the finite element simulation results corresponding to the design parameters.

[0007] As a further solution of the present invention, the process of establishing the pressure vessel geometric model is as follows: First, determine the material, geometric model and finite element simulation parameters of the pressure vessel, and select Design parameters form a vector , and each set of vectors Constitute a set of training samples; Indicates the Design parameters, , Transpose flag for vector; Afterwards, according to the vector The geometric model of the pressure vessel is established based on the design parameters in .

[0008] As a further solution of the present invention: the calculation process of the finite element simulation results in step 1 is as follows: First, select Group training samples , and calculate the finite element simulation results of the pressure vessel geometric model corresponding to each group of training samples, namely the node displacement results; the corresponding node displacement results are , the node displacement result is the node displacement vector formed by arranging the displacement values ​​of each node in each direction in sequence; Indicates the Group training samples, express The corresponding node displacement results are, ; Next, the node displacement results corresponding to each group of training samples are stored in the current group of training samples; Finally, select from all training samples The group is used as the initial training sample and selected from the remaining text group as test samples, and there are .

[0009] As a further solution of the present invention, the process of constructing the design parameter space is as follows: First, construct a dimensional reduced basis space , Indicates the The finite element simulation results corresponding to the training samples are ; Next, Schmidt orthogonalization is performed on the reduced basis space to obtain the orthogonal space basis vectors , Represents the spatial dimension and is calculated separately through the spatial basis vectors The reduced basis solution corresponding to the initial training samples of the group is: ; Among them, vector ,express The reduced basis solution of ; Finally, each reduced basis solution is combined in sequence to form a reduced basis solution space , and create the selected The design parameter space corresponding to the initial training samples of the group , Indicates the Set initial training samples; As a further solution of the present invention: the process of establishing the Gaussian process regression model is as follows: First, create a The training set of the initial training samples: ; in, To reduce the base solution A dataset of components, for No. A quantity, , Indicates the total number of components; Next, set the fitting function to the autocorrelation-determined squared exponential kernel function: ; in, is the standard deviation, For the The characteristic length of the design parameter, for The prediction results, and They are and No. Design parameters; Indicates that the kernel function is and The matrix calculated above; thus the covariance matrix is ​​established as: ; Where, represents the covariance matrix; Indicates that the kernel function is in the design parameter space The matrix calculated above; represents the noise variance; express -dimensional identity matrix; Finally, the relevant parameters of the kernel function are optimized by maximum likelihood estimation to fit the Gaussian process regression model. , and based on Establish a vector consisting of the reduced basis solution and design parameters The nonlinear mapping relationship between: ; Where, represents the first A portion.

[0010] As a further solution of the present invention, the process of using the Gaussian process regression model to obtain the predicted solution of the finite element simulation results is as follows: First, select from the remaining training samples The training samples that are not used to fit the Gaussian process regression model are used as test samples. , Indicates the Group test samples, ; The corresponding reduced basis prediction solution is calculated using the Gaussian process regression model: ; Where, express The reduced basis prediction solution Quantity express The reduced basis prediction solution of ; Then, the reduced basis prediction solution is mapped back to the finite element space, and the prediction solution of the finite element simulation result, that is, the node displacement result, is obtained as follows: ; Where, express Predicted solution of nodal displacement results obtained through mapping.

[0011] As a further solution of the present invention, the training process of the Gaussian process regression model is as follows: First, the relative error between the predicted solution and the true value of the node displacement result is calculated ; Then, compare the relative error with the set convergence tolerance If the size relationship between , then the Gaussian process regression model converges, indicating that the Gaussian process regression model has completed training; if , then construct a set of design parameters and let , loop through steps 2 to 4 until .

[0012] As a further solution of the present invention: the relative error calculation formula is as follows: ; Where, represents the second-order norm; Indicates the The true value of the test sample.

[0013] As a further solution of the present invention, the process of using the Gaussian process regression model to predict the finite element simulation results corresponding to the design parameters is as follows: For any new set of design parameters consisting of a vector , using the trained Gaussian process regression model, calculate the corresponding reduced basis solution, and map the corresponding node displacement result prediction solution ; Then, according to the basic finite element theory, the unit strain matrix is ​​calculated , calculate the strain of the cell or Gauss point , and substituted into the corresponding constitutive calculation method, we get the stress and equivalent plastic strain Finite element simulation results of pressure vessels including.

[0014] As a further solution of the present invention: predicting the solution The calculation formula is as follows: ; Where, Represents a vector Reduced basis solution computed via a Gaussian process regression model.

[0015] Compared with the prior art, the present invention has the following beneficial effects: 1. The present invention improves the finite element method by adopting the reduced basis method, and develops a solution method that combines online and offline methods. After presetting design variables such as structural dimensions, load type and size, and material properties, offline solution is first used to obtain the reduced basis space, and then a mapping relationship is established between the reduced basis solution and the finite element solution. This reduces the degree of freedom of the finite element method in solving mechanical problems, greatly reduces the amount of calculation required for the traditional finite element method to solve a system of linear algebraic equations while ensuring accuracy, and can achieve a rapid solution to the deformation behavior of pressure vessels.

[0016] 2. The present invention aims at the elastic-plastic deformation behavior of pressure vessels, introduces the Gaussian process regression method into the reduced basis finite element method to establish the connection between the design parameter space and the reduced basis space, and replaces the nonlinear iterative method for elastic-plastic deformation in the finite element method. During offline solution, the regression model of the reduced basis space and the design parameter space is fitted by the Gaussian process regression method, and the mapping relationship between the design parameter space and the node displacement solution space is further established in combination with the reduced basis method, so that the node displacement results of different design parameters are obtained through model mapping and low-degree-of-freedom matrix operations during the online calculation process. Compared with the traditional finite element method, this method avoids the solution and iteration of large linear algebraic equations in the online stage, and the solution speed is increased by more than 50 times. The error between the simulated deformation result and the traditional finite element method is only 1%, which can be used for fast and real-time simulation of specific equipment such as pressure vessels under various working conditions.

[0017] 3. The present invention establishes an iterative solution format based on error analysis during offline solution. By performing trial calculations and error analysis on test samples, the number of training samples for the reduced basis method and Gaussian process regression model is iteratively gradually increased. This can minimize the number of samples and calculation time for offline calculations, as well as the degrees of freedom for online calculations, while ensuring solution accuracy, and avoid overfitting due to excessive training samples. The computational degrees of freedom of the present invention are the same as the number of training samples. Through simple calculations and mapping, the numerical simulation results corresponding to different design parameters can be quickly and accurately simulated. In addition to simulating elastic-plastic problems, the method can also be used for other material nonlinearities, geometric and contact nonlinearities, such as fatigue and damage. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a technical flow chart of the present invention.

[0019] Figure 2 This is the geometric model of Example 1 of the present invention.

[0020] Figure 3 This is the grid division situation of Example 1 of the present invention.

[0021] Figure 4 It is the iterative error curve of Example 1.

[0022] Figure 5 This is the displacement result calculated using the method of the present invention in Example 1.

[0023] Figure 6 The displacement results calculated by the finite element method in Example 1 are shown in FIG.

[0024] Figure 7 This is the Mises stress result calculated using the method of the present invention in Example 1.

[0025] Figure 8 This is the Mises stress result calculated using the finite element method in Example 1.

[0026] Figure 9 This is the equivalent plastic strain result calculated by the method of the present invention in Example 1.

[0027] Figure 10 The equivalent plastic strain result calculated by the finite element method in Example 1 is shown in FIG.

[0028] Figure 11 This is the geometric model and mesh division of Example 2 of the present invention.

[0029] Figure 12 This is the displacement result calculated by the present invention corresponding to the first set of design parameters in Example 2 of the present invention.

[0030] Figure 13 This is the displacement result calculated by the finite element method corresponding to the first set of design parameters in Example 2 of the present invention.

[0031] Figure 14 This is the Mises stress result calculated by the present invention corresponding to the first set of design parameters in Example 2 of the present invention.

[0032] Figure 15 This is the Mises stress result calculated by the finite element method corresponding to the first set of design parameters in Example 2 of the present invention. DETAILED DESCRIPTION

[0033] The following will clearly and completely describe the technical solutions and implementation processes of the present invention in conjunction with the embodiments of the present invention and the accompanying drawings. Obviously, the described embodiments are only used to illustrate the implementation process and effects of the present invention and do not limit the scope of application of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0034] Figure 1 To more clearly demonstrate the implementation objectives, solutions, and specific implementation effects of the present invention, the following further illustrates the accuracy, applicability, and other excellent performance of the intelligent reduced basis finite element method proposed by the present invention through several examples and accompanying drawings.

[0035] 1. Pressure vessel geometric model A geometric model of the pressure vessel is established for finite element simulation, and this model is used to calculate the corresponding finite element simulation results for each set of design parameters. Building a parametric geometric model is the foundation of the method implementation, ensuring that the model can dynamically respond to changes in design parameters and providing a unified framework for subsequent order reduction calculations.

[0036] 1. Design parameter definition Select the key parameters that affect the deformation behavior of the pressure vessel, such as the material, geometric model and finite element simulation parameters of the pressure vessel. Design parameters form a vector , and each set of vectors Construct a set of training samples to avoid the efficiency loss of repeated modeling. Indicates the Design parameters, , Transpose flag for vectors.

[0037] 2. Establish the geometric model of pressure vessel In the intelligent reduced-base finite element method for pressure vessel deformation analysis, finite element software, such as ANSYS, ABAQUS and other finite element analysis software, as the core technology carrier, plays the dual role of key solver and engineering implementation platform. Based on finite element analysis software and according to vector The geometric model of the pressure vessel is established based on the design parameters in Figure 2 and Figure 3 .

[0038] 3. Finite element calculation Select Group training samples , where each vector is arranged in sequence according to the node direction. The finite element simulation results of the pressure vessel geometric model corresponding to each group of training samples, namely the node displacement results, are calculated respectively. The corresponding node displacement results are , the node displacement result is the node displacement vector formed by arranging the displacement values ​​of each node in each direction in sequence; Indicates the Group training samples, express The corresponding node displacement results are, .

[0039] Then, the node displacement results corresponding to each group of training samples are stored in the current group of training samples.

[0040] Finally, select from all training samples The group is used as the initial training sample and selected from the remaining text group as test samples, and there are .

[0041] 2. Design parameter space In order to solve the computational bottleneck of high-dimensional finite element equations, an order reduction strategy is adopted. The reduced basis space is constructed through the finite element simulation results, and the reduced basis space is Schmidt orthogonalized to obtain the orthogonal space basis vectors; the reduced basis solution is calculated through the orthogonal space basis vectors, and the reduced basis solution space is constructed. At the same time, the design parameter space is constructed based on the training samples.

[0042] 1. Reduce base space In order to extract the core deformation mode of the pressure vessel (such as the axisymmetric expansion of the pressure vessel), it is necessary to eliminate redundant modes, so the key mode is concentrated by reducing the basis space. That is, a dimensional reduced basis space , Indicates the The node displacement results corresponding to the training samples are: .

[0043] 2. Reduced base solution Perform Schmidt orthogonalization on the reduced basis space to avoid the ill-conditioned matrix problem and obtain the orthogonal space basis vectors , Represents the spatial dimension and is calculated separately through the spatial basis vectors The reduced basis solution corresponding to the initial training samples of the group is: (1); Among them, vector ,express The reduced basis solution of .

[0044] 3. Design parameter space Combine each reduced basis solution in sequence to form a reduced basis solution space , and create the selected The design parameter space corresponding to the initial training samples of the group , Indicates the A set of initial training samples.

[0045] As a physical intelligence carrier, it condenses key deformation modes from finite element sample solutions and compresses the ten-thousand-dimensional degrees of freedom into a hundred-dimensional reduced-order space by constructing an orthogonal basis vector matrix through Schmidt orthogonalization. This not only filters numerical noise but also directly calls the orthogonal space basis vector mapping (Formula (9)) during the online prediction phase, significantly reducing the grid model calculation time (for example, from 46 seconds to 1.64 seconds in Table 2) while strictly ensuring the physical rationality of the solution. Its adaptive evolution capability, namely updating the orthogonal space basis vectors as samples increase, further supports real-time simulation of complex scenarios and provides the underlying technical foundation for real-time industrial monitoring and optimization design.

[0046] 3. Gaussian Process Regression Model The Gaussian process regression method is used to establish the mapping relationship between the design parameter space and the reduced basis solution space. At the same time, the auto-correlation square exponential kernel function is selected as the kernel function of the Gaussian regression process. The kernel function is optimized through parameter optimization using maximum likelihood estimation. Based on the optimized kernel function, a Gaussian process regression model is established to characterize the nonlinear mapping relationship between the reduced basis solution and the design parameters.

[0047] 1. Component modeling In order to avoid overfitting caused by high-dimensional output, component modeling is adopted: (2); in, To reduce the base solution A dataset of components, for No. A quantity, , Indicates the total number of components.

[0048] 2. Kernel Function The auto-correlation-determined squared exponential kernel function adaptively quantifies the differential effects of different design parameters (such as wall thickness and internal pressure) on pressure vessel deformation, automatically identifying key sensitive parameters through characteristic length parameters. Furthermore, its infinitely differentiable nature allows it to accurately fit the strong nonlinear relationships of elastic-plastic deformation. While ensuring minimal displacement prediction error, it requires only hundreds of training samples to achieve convergence and enables millisecond-level online prediction. This method perfectly resolves the multiple conflicts between traditional methods in identifying parameter sensitivity, nonlinear modeling, and real-time performance. Therefore, the present invention uses the auto-correlation-determined squared exponential kernel function as the kernel function for the Gaussian regression method.

[0049] (3); Where, For the Characteristic length of various design parameters. Each design parameter (such as wall thickness, internal pressure) has an independent characteristic length. , thereby automatically quantifying the influence weight of design parameters on deformation. is the standard deviation. for The prediction results, and They are and No. Design parameters; Indicates that the kernel function is and The matrix calculated above. The covariance matrix is ​​established as: (4); Where, represents the covariance matrix; Indicates that the kernel function is in the design parameter space The matrix calculated above; represents the noise variance; express -dimensional identity matrix.

[0050] 3. Maximum Likelihood Estimation Optimization Maximum likelihood estimation can automatically balance model accuracy and generalization ability based on the principles of probability statistics: this method maximizes the likelihood function of the training samples, allowing the Gaussian process regression model to accurately capture the nonlinear relationship between design parameters and deformation responses, while suppressing the numerical perturbations of finite element calculations through noise terms to avoid overfitting. Compared with traditional optimization methods such as grid search, maximum likelihood estimation can converge efficiently with hundreds of samples (for example, only 124 groups of samples are required in Example 1), and the optimized kernel function parameters have clear physical meanings, providing a quantitative basis for engineering decisions. Therefore, the relevant parameters of the kernel function are optimized through maximum likelihood estimation, and the Gaussian process regression model is fitted. , and based on Establish a vector consisting of the reduced basis solution and design parameters The nonlinear mapping relationship between: (5); Where, represents the first A portion.

[0051] 4. Train the Gaussian process regression model through the finite element simulation results until the prediction accuracy requirements are met; and use the trained Gaussian process regression model to predict the finite element simulation results corresponding to the design parameters.

[0052] 1. Use Gaussian process regression model to obtain finite element simulation results First, select from the remaining training samples The training samples that are not used to fit the Gaussian process regression model are used as test samples. , Indicates the Group test samples, .

[0053] The corresponding reduced basis prediction solution is calculated using the Gaussian process regression model: (6); Where, express The reduced basis prediction solution Quantity express The reduced basis prediction solution of .

[0054] Then, the reduced basis prediction solution is mapped back to the finite element space, and the prediction solution of the finite element simulation result, that is, the node displacement result, is obtained as follows: (7); Where, express Predicted solution of nodal displacement results obtained through mapping.

[0055] 2. Training Gaussian Process Regression Model First, the relative error between the predicted solution and the true value of the node displacement result is calculated : (8); Where, represents the second-order norm; Indicates the The true value of the test sample.

[0056] Then, compare the relative error with the set convergence tolerance If the size relationship between , then the Gaussian process regression model converges, indicating that the Gaussian process regression model has completed training; if , then construct a set of design parameters and let , loop through steps 2 to 4 until .

[0057] 3. Use Gaussian process regression model to predict finite element simulation results For any new set of design parameters consisting of a vector , using the trained Gaussian process regression model, calculate the corresponding reduced basis solution, and map the corresponding node displacement result prediction solution : Prediction Solution The calculation formula is as follows: (9); Where, Represents a vector Reduced basis solution computed via a Gaussian process regression model.

[0058] Then, according to the basic finite element theory, the unit strain matrix is ​​calculated , calculate the strain of the cell or Gauss point , and substituted into the corresponding constitutive calculation method, we get the stress and equivalent plastic strain Finite element simulation results of pressure vessels including.

[0059] Through strict closed-loop verification and efficient online prediction mechanism, a double breakthrough in the accuracy and efficiency of pressure vessel deformation analysis was achieved: First, based on the error feedback of the test sample (Formula (8)), the adaptive optimization model was used to reduce the displacement prediction error to 0.58% (Example 1). At the same time, the online calculation complexity was compressed through orthogonal basis mapping (Formula (9)), and the response speed of the 10,000-level grid model was greatly improved; secondly, the complete strain / stress field output ( Figures 7 to 10 ) ensures engineering practicality, while the dynamic sample expansion strategy reduces redundant calculations by 80%. Ultimately, while maintaining 99% accuracy for elastic-plastic problems, real-time simulation is achieved in seconds (Example 2 takes only 2.36 seconds), providing a highly efficient tool for pressure vessel design optimization and safety monitoring.

[0060] Five, embodiment: 1. Example 1: Rapid prediction of elastic-plastic deformation behavior of cylinders of different sizes under internal and external pressure: This example considers the elastic-plastic deformation behavior of a typical pressure vessel structure, a thick-walled cylinder, under internal and external pressure. Its geometry is as follows: Figure 2 As shown, the inner and outer sides are subjected to pressure respectively and The relevant parameters of the action, cylinder length and bilinear elastic-plastic material are shown in Table 1.

[0061] Table 1 Model parameters and material properties of Example 1 ; In order to calculate the deformation and stress of cylinders of different sizes under different pressure loads, the method proposed in this invention is used for rapid simulation. First, the design variables are determined as , where the outer radius of the cylinder is , cylinder thickness , . Establish the finite element model as Figure 2 As shown in , the model is divided into 4000 units and 5040 nodes. During simulation, the axial displacement of the end face is constrained, as shown in Figure 3 shown.

[0062] Then, the offline mode calculation of this embodiment is carried out to define the initial training sample number , the number of test samples , given within the parameter range of the design variables The design variables are combined into sample parameters, the finite element results are calculated and 5 groups of samples are randomly selected to construct the reduced basis space. , as training samples, and Schmidt orthogonalization is performed to obtain , calculate the reduced basis solution corresponding to each group of samples according to formula (1) , refer to formulas (2) to (5) and establish the Gaussian process regression method and The mapping relationship.

[0063] Then refer to formulas (6) to (7) to predict the node displacement results of another 10 groups of test samples , and refer to the corresponding finite element solution, calculate the error according to formula (8), if the error is greater than , then add a set of sample parameters and calculate the finite element solution, and then increase the number of training samples accordingly , reconstruct the reduced basis space until the error meets the convergence requirements.

[0064] Finally, the mapping relationship between the trained reduced basis space and Gaussian process regression can be used to carry out online prediction for any parameter The corresponding reduced basis solution can be obtained by mapping , and then calculate the node displacement results according to formula (9), and substitute them into the constitutive relationship to calculate the stress and strain. This online prediction process does not require solving a large-scale linear algebraic equation system when calculating the node displacement results. It only needs to perform matrix multiplication operations, and then calculate the strain and stress based on the node displacement results. This can greatly improve the solution efficiency. In addition, because the finite element solution is used as the reference solution for model reduction in the offline stage, the accuracy of the results is guaranteed.

[0065] When the offline calculation converges in this embodiment, the number of training samples used is 124. The relationship between the test error and the number of samples in the offline calculation process is as follows: Figure 4 As shown. The trained pressure vessel geometry model is then used to predict , , and The node displacement results are 220 and 110 MPa respectively, and the strain and stress are calculated. In order to give the stress and strain state of the material more intuitively, the Mises equivalent stress of the element is calculated. and equivalent plastic strain as follows: (10); (11); Where, the deviatoric stress tensor is , is a second-order unit tensor, is the plastic strain tensor, is the Cauchy stress tensor. Figures 5 to 10The results of the structural displacement, Mises equivalent stress and equivalent plastic strain distribution under the prediction parameters of this embodiment obtained by the present invention and finite element calculation are given. Further calculation according to formula (8) shows that the node displacement error between the present invention and the finite element solution is only 0.58%. In order to further illustrate the acceleration effect of the present invention, the grid size is increased and decreased respectively, the model is trained, and the calculation time is obtained as shown in Table 2. It can be seen that the calculation time of the intelligent reduced basis finite element method proposed by the present invention is basically unchanged, while the finite element method is greatly affected by the number of grids. This is because the time consumed by the online calculation of the present invention is mainly used for the calculation of unit strain and stress, and only matrix multiplication operations are required when solving displacements; while the stiffness matrix needs to be inverted when solving the linear algebraic equations by finite elements, and the dimension of the reduced basis matrix is ​​only proportional to the number of training samples. r related to The stiffness matrix of the finite element method is related to the displacement degrees of freedom. Therefore, the more degrees of freedom the nodes in the pressure vessel geometry model have, the more significant the acceleration effect of the present invention compared to the traditional finite element method, which demonstrates the rapid calculation effect of the present invention.

[0066] Table 2 Calculation time of Example 1 using the present invention and traditional FEM ; This embodiment simulates and analyzes the elastic-plastic mechanical behavior of a thick-walled cylinder, a typical structure of a pressure vessel, under internal and external pressures. The radius, thickness, and internal and external pressures of the thick-walled cylinder are used as design variables. The accuracy and rapid calculation effect of the present invention are verified, indicating the potential of the present invention in the rapid simulation design of pressure vessels.

[0067] 2. Example 2: Rapid prediction of elastic-plastic deformation behavior of horizontal pressure vessels under different loads: This embodiment considers the deformation of a horizontal pressure vessel structure under parameters such as internal pressure, gravity, and seismic load. The specific implementation is similar to that of Example 1 and will not be repeated here. The model and mesh division of this embodiment are as follows: Figure 11 As shown, the model is divided into 65,776 elements and 81,546 nodes, with four mesh layers along the tank thickness. To more clearly illustrate the internal structure, only half of the model is shown. The materials for both the saddle and the tank are set to bilinear elastoplastic. The relevant mechanical properties are shown in Table 3.

[0068] Table 3 Material parameters of horizontal storage tanks ; Table 4 shows the main geometric parameters of the tank structure.

[0069] Table 4 Main structural parameters of horizontal storage tanks ; During the simulation, the bottom end of the tank saddle is fixed, and the load conditions that need to be considered in the design and use of the tank are considered, and the uniformly distributed pressure on the tank is considered. , the whole is subject to gravity And the earthquake acceleration in three directions 、 and The role of design parameters ,in , 、 and m / s 2 , The size is 9.8m / s 2 , the direction is Negative direction of the axis.

[0070] This embodiment first performs offline mode calculations, defines the initial number of training samples as 5 and the number of test samples as 10. The prediction model converged when , and the average node displacement error obtained from the calculated test set was 0.4%. The converged model was used to calculate three models with different design parameters. The design parameters, simulation errors, and calculation time are shown in Table 5.

[0071] Table 5 Comparison of main structural parameters and calculation results of horizontal storage tanks ; The stress and displacement results obtained by the present invention and the finite element method under three conditions are further given. The results of the first set of design parameters are as follows: Figures 12 to 15 As shown in the figure, similar to the previous one, only half of the model is shown to more clearly show the displacement and stress inside the model. Figure 14 、 Figure 15 The results from the material parameters show that for the first set of design parameters, the tank is in the elastic phase and does not undergo plastic deformation. However, for the second and third sets of load parameters, which result in higher internal pressures, the tank undergoes varying degrees of plastic deformation. The results of the present invention and the finite element method demonstrate that the present invention can quickly and accurately predict the stress and deformation of the tank for all three scenarios.

[0072] In this embodiment, the model obtained after offline calculation can use online calculation to quickly obtain simulation results for each different design parameter, and the time spent is only 1% of the finite element method, and the error between the two is only 1%, which proves that the present invention can quickly and accurately simulate the elastic-plastic deformation behavior of pressure vessels under loads such as earthquakes and internal pressure.

[0073] The simulations of two examples demonstrate the excellent performance of the present invention in rapidly simulating the elastic-plastic deformation of pressure vessels, demonstrating its significant potential for real-time rapid simulation of pressure vessels. The present invention can rapidly generate simulation results for different geometric and load parameters during the pressure vessel design process, increasing the efficiency of pressure vessel design. It can also rapidly simulate the deformation and stress state of pressure vessels in real time based on real-time monitored load data.

[0074] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. An intelligent reduced basis finite element method for rapid analysis of pressure vessel deformation behavior, characterized in that: The following steps are involved: Step 1: Establish a pressure vessel geometry model for finite element simulation, and use the pressure vessel geometry model to calculate the finite element simulation results corresponding to each set of design parameters; Step 2: Construct a reduced basis space based on the finite element simulation results, perform Schmidt orthogonalization on the reduced basis space to obtain orthogonal space basis vectors; calculate the reduced basis solution through the orthogonal space basis vectors, construct the reduced basis solution space, and construct the design parameter space based on the training samples; Step 3: Use the Gaussian process regression method to establish a mapping relationship between the design parameter space and the reduced basis solution space. At the same time, select the square exponential kernel function determined by auto-correlation as the kernel function of the Gaussian regression process. Use maximum likelihood estimation to optimize the kernel function through parameter optimization. Based on the optimized kernel function, establish a Gaussian process regression model that characterizes the nonlinear mapping relationship between the reduced basis solution and the design parameters. Step 4: Train the Gaussian process regression model using finite element simulation results until the prediction accuracy requirements are met; The trained Gaussian process regression model is then used to predict the finite element simulation results corresponding to the design parameters.

2. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 1, characterized in that: The process of establishing the pressure vessel geometric model is as follows: First, determine the material, geometric model and finite element simulation parameters of the pressure vessel, and select Design parameters form a vector , and each set of vectors Constitute a set of training samples; Indicates the Design parameters, , Transpose flag for vector; Afterwards, according to the vector The geometric model of the pressure vessel is established based on the design parameters in .

3. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 2, characterized in that: The calculation process of the finite element simulation results in step 1 is as follows: First, select Group training samples , and calculate the finite element simulation results of the pressure vessel geometric model corresponding to each group of training samples, namely the node displacement results; the corresponding node displacement results are , the node displacement result is the node displacement vector formed by arranging the displacement values ​​of each node in each direction in sequence; Indicates the Group training samples, express The corresponding node displacement results are, ; Next, the node displacement results corresponding to each group of training samples are stored in the current group of training samples; Finally, select from all training samples The group is used as the initial training sample and selected from the remaining text group as test samples, and there are .

4. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 3, characterized in that: The process of constructing the design parameter space is as follows: First, construct a dimensional reduced basis space , Indicates the The node displacement results corresponding to the training samples are: ; Next, Schmidt orthogonalization is performed on the reduced basis space to obtain the orthogonal space basis vectors , Represents the spatial dimension and is calculated separately through the spatial basis vectors The reduced basis solution corresponding to the initial training samples of the group is: ； Among them, vector ,express The reduced basis solution of ; Finally, each reduced basis solution is combined in sequence to form a reduced basis solution space , and create the selected The design parameter space corresponding to the initial training samples of the group , Indicates the A set of initial training samples.

5. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 4, characterized in that: The process of establishing the Gaussian process regression model is as follows: First, create a The training set of the initial training samples: ； in, To reduce the base solution A dataset of components, for No. A quantity, , Indicates the total number of components; Next, set the fitting function to the autocorrelation determined square exponential kernel function; ； Where, is the standard deviation, For the The characteristic length of the design parameter, for The prediction results, and They are and No. Design parameters; Indicates that the kernel function is and The matrix calculated above; Finally, the relevant parameters of the kernel function are optimized by maximum likelihood estimation, and the Gaussian process regression model is fitted. , and based on Establish a vector consisting of the reduced basis solution and design parameters The nonlinear mapping relationship between: ； Where, represents the first A portion.

6. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 5, characterized in that: The process of using the Gaussian process regression model to obtain a predicted solution for finite element simulation results is as follows: First, select from the remaining training samples The training samples that are not used to fit the Gaussian process regression model are used as test samples. , Indicates the Group test samples, ; The corresponding reduced basis prediction solution is calculated using the Gaussian process regression model: ； Where, express The reduced basis prediction solution Quantity express The reduced basis prediction solution of ; Then, the reduced basis prediction solution is mapped back to the finite element space, and the prediction solution of the finite element simulation result, that is, the node displacement result, is obtained as follows: ； Where, express Predicted solution of nodal displacement results obtained through mapping.

7. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 6, characterized in that: The training process of the Gaussian process regression model is as follows: First, the relative error between the predicted solution and the true value of the node displacement result is calculated ; Then, compare the relative error with the set convergence tolerance If the size relationship between , then the Gaussian process regression model converges, indicating that the Gaussian process regression model has completed training; if , then construct a set of design parameters and let , loop through steps 2 to 4 until .

8. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 7, characterized in that: The relative error calculation formula is as follows: ； Where, represents the second-order norm; Indicates the The true value of the test sample.

9. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 8, characterized in that: The process of using the Gaussian process regression model to predict the finite element simulation results corresponding to the design parameters is as follows: For any new set of design parameters consisting of a vector , using the trained Gaussian process regression model, calculate the corresponding reduced basis solution, and map the corresponding node displacement result prediction solution ; Then, according to the basic finite element theory, the unit strain matrix is ​​calculated , calculate the strain of the cell or Gauss point , and substituted into the corresponding constitutive calculation method, we get the stress and equivalent plastic strain Finite element simulation results of pressure vessels including.

10. The intelligent reduced basis finite element method for rapid analysis of deformation behavior of pressure vessels according to claim 9, characterized in that: Prediction Solution The calculation formula is as follows: ； Where, Represents a vector Reduced basis solution computed via a Gaussian process regression model.

Images