Intelligent reduced basis finite element method for fast analysis of deformation behavior of pressure vessels

Abstract

The application belongs to the technical field of pressure vessel structure analysis, and specifically relates to an intelligent reduced basis finite element method for fast analysis of pressure vessel deformation behavior. The application improves the finite element method by using the reduced basis method, and develops an online and offline combined solving mode. After design parameters such as structure size, load type and size, material properties and the like are preset, the reduced basis space is obtained by using the offline solving mode, and then a mapping relationship between the reduced basis solution and the finite element solution is established, so that the finite element model is reduced, the degree of freedom of the problem is reduced, the calculation amount required for solving the linear algebraic equation set by the traditional finite element method is greatly reduced while the accuracy is ensured, and further, the Gaussian process regression method is introduced to describe the elastic-plastic deformation behavior of the pressure vessel, so that the fast solution of the pressure vessel deformation behavior is realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of pressure vessel structure analysis, and specifically relates to an intelligent reduced basis finite element method for rapid analysis of deformation behavior of a pressure vessel. BACKGROUND

[0002] A pressure vessel refers to a container device capable of bearing internal pressure load, and is widely used in various important industrial fields such as petroleum and chemical industry, nuclear industry, aerospace, etc. Since the pressure vessel often contains flammable and explosive hazardous materials, and is subjected to pressure, self-weight, nozzle load and other loads during operation, failure behavior may cause great danger. Therefore, the stress state and deformation behavior under load are of great significance to the design, safety assessment and operation and maintenance of the pressure vessel. 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 the pressure vessel.

[0003] The finite element analysis method is often used for structural design and safety analysis of pressure vessels. In the process of structural design optimization, the design parameters of the pressure vessel need to be changed for specific targets and simulation analysis is performed to find the optimal design scheme. In addition, during the operation of large pressure vessels, the finite element method can also be used to simulate and analyze the deformation behavior and stress state of the pressure vessel under certain loads. In the process of pressure vessel optimization design and operation simulation, the related design variables or load parameters need to be changed repeatedly to perform finite element analysis on the structure, which consumes a large amount of computing resources and time. The calculation speed of the finite element method often depends on the number of meshes and nodes. The analysis of a pressure vessel model with a large number of meshes requires a large amount of calculation and a lot of calculation time. In addition, the pressure vessel may exhibit nonlinear elastic-plastic deformation behavior during long-term service or under heavy load, which requires iterative solution for numerical analysis, further increasing the calculation amount. It can be seen that numerical methods such as finite element method still have great limitations in the design and analysis of complex structure pressure vessels, especially in real-time deformation behavior and stress state monitoring analysis. Therefore, it is necessary to develop an efficient numerical analysis method to reduce the simulation calculation time required for changing the structure or load parameters each time, so as to realize the rapid simulation analysis of the mechanical behavior of the pressure vessel.

[0004] The method for improving the calculation efficiency of the finite element algorithm can be considered from two aspects of improving computer performance and reducing the degree of freedom of calculation. The method for improving computer performance needs too high cost and is too dependent on the development of computer hardware technology. The method for reducing the degree of freedom by generally increasing the grid size reduces the calculation accuracy, thereby reducing the reliability of the calculation result. The reduced basis method (RBM) maps the high-dimensional solution space to the low-dimensional space through the Galerkin method, thereby greatly reducing the dimension and calculation amount of the linear algebraic equation set, and ensuring the accuracy of the result through a reliable space mapping error estimation method, thereby realizing accurate and fast solving of the equation set. The RBM has great application potential for problems in which the control equation is unchanged, and the physical parameters, the solving region and the boundary condition are constantly changed. However, in the general RBM, the solution space and the low-dimensional space are connected through linear mapping, and it is difficult to be used for solving the nonlinear problems such as elastic-plastic deformation commonly used in pressure vessels. SUMMARY

[0005] In order to avoid and overcome the technical problems existing in the prior art, the present application provides an intelligent reduced basis finite element method for fast analysis of deformation behavior of a pressure vessel. The present application effectively reduces the calculation complexity, thereby effectively improving the generation efficiency of the finite element simulation result.

[0006] To achieve the above object, the present application provides the following technical scheme.

[0007] An intelligent reduced basis finite element method for fast analysis of deformation behavior of a pressure vessel comprises the following steps:

[0008] Step 1: establishing a pressure vessel geometric model for finite element simulation, and using the pressure vessel geometric model to calculate the finite element simulation result corresponding to each group of design parameters;

[0009] Step 2: constructing a reduced basis space through the finite element simulation result, obtaining orthogonal space basis vectors by Schmidt orthogonalization of the reduced basis space, calculating the reduced basis solution through the orthogonal space basis vectors, constructing a reduced basis solution space, and constructing a design parameter space based on the training sample;

[0010] Step 3: establishing a mapping relationship between the design parameter space and the reduced basis solution space by using a Gaussian process regression method, selecting an automatic correlation determination square exponential kernel function as the kernel function of the Gaussian regression process, optimizing the kernel function by parameter optimization through maximum likelihood estimation, and establishing a Gaussian process regression model representing the nonlinear mapping relationship between the reduced basis solution and the design parameter based on the optimized kernel function;

[0011] Step 4: training the Gaussian process regression model through the finite element simulation result until the prediction accuracy requirement is met, and predicting the finite element simulation result corresponding to the design parameter by using the trained Gaussian process regression model.

[0012] As a further solution of the present invention, the process of establishing the pressure vessel geometric model is as follows:

[0013] 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;

[0014] Afterwards, according to the vector The geometric model of the pressure vessel is established based on the design parameters in .

[0015] As a further solution of the present invention: the calculation process of the finite element simulation results in step 1 is as follows:

[0016] 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, ;

[0017] Next, the node displacement results corresponding to each group of training samples are stored in the current group of training samples;

[0018] 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 .

[0019] As a further solution of the present invention, the process of constructing the design parameter space is as follows:

[0020] First, construct a dimensional reduced basis space , Indicates the The finite element simulation results corresponding to the training samples are ;

[0021] Then, the Schmidt orthogonalization is performed on the reduced basis space to obtain orthogonal space basis vectors , represents the space dimension, and is calculated by the space basis vectors respectively The reduced basis solution corresponding to the initial training sample set is:

[0022] ;

[0023] wherein, the vector represents the reduced basis solution of ;

[0024] Finally, the reduced basis solutions are combined in sequence to form a reduced basis solution space , and the design parameter space corresponding to the selected initial training sample set is established , represents the initial training sample set No.

[0025] As a further scheme of the present application: the establishment process of the Gaussian process regression model is as follows:

[0026] Firstly, a training set containing initial training sample set is established:

[0027] ;

[0028] wherein, is the data set of the first component of the reduced basis solution, is the first component of , , , , represents the total number of components;

[0029] Then, the fitting function is set as the automatic correlation determination square exponential kernel function:

[0030] ;

[0031] wherein, is the standard deviation, is the characteristic length of the design parameter No. , is the prediction result of , and are the design parameter No. and of respectively; represents the kernel function in and ​The matrix calculated above; thereby the covariance matrix is established as:

[0032] ;

[0033] wherein, denotes the covariance matrix; denotes the kernel function in the design parameter space The matrix calculated above; denotes the noise variance; denotes the unit matrix of dimension

[0034] Finally, the relevant parameters of the kernel function are optimized by maximum likelihood estimation, and the Gaussian process regression model is fitted , and the nonlinear mapping relationship between the reduced basis solution and the vector composed of the design parameters is established :

[0035] ;

[0036] wherein, denotes the first component of the reduced basis solution of the th test sample.

[0037] As a further scheme of the present application: the process of using the Gaussian process regression model to obtain the prediction solution of the finite element simulation result is as follows:

[0038] First, a group of training samples that are not used to fit the Gaussian process regression model are selected from the remaining training samples as test samples , denotes the th test sample, ;

[0039] The corresponding reduced basis prediction solution is calculated by using the Gaussian process regression model:

[0040] ;

[0041] wherein, denotes the first component of the reduced basis prediction solution of the th test sample; denotes the reduced basis prediction solution of the th test sample; Then, the reduced basis prediction solution is mapped back to the finite element space, and the prediction solution of the finite element simulation result, i.e., the node displacement result, is obtained as:

[0042]

[0043] ;

[0044] wherein,​​​ express Predicted solution of nodal displacement results obtained through mapping.

[0045] As a further solution of the present invention, the training process of the Gaussian process regression model is as follows:

[0046] First, the relative error between the predicted solution and the true value of the node displacement result is calculated ;

[0047] 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 .

[0048] As a further solution of the present invention: the relative error calculation formula is as follows:

[0049] ;

[0050] Where, represents the second-order norm; Indicates the The true value of the test sample.

[0051] 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:

[0052] 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 ;

[0053] 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.

[0054] As a further solution of the present invention: predicting the solution The calculation formula is as follows:

[0055] ;

[0056] Where, representative vector Reduced basis solution calculated by Gaussian process regression model.

[0057] Compared with the prior art, the present application has the beneficial effects that:

[0058] 1. The present application improves the finite element method by using the reduced basis method, and develops an online and offline combined solution method. After the design variables such as structure size, load type and size, material properties, etc. are pre-set, the reduced basis space is obtained by offline solution, and the mapping relationship between the reduced basis solution and the finite element solution is established, thereby reducing the degree of freedom of the finite element method for solving mechanical problems, greatly reducing the calculation amount required for solving linear algebraic equations in the traditional finite element method while ensuring accuracy, and realizing fast solution of the deformation behavior of the pressure vessel.

[0059] 2. The present application introduces a 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, instead of the nonlinear iteration method for elastic-plastic deformation in the finite element method. In 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 by combining the reduced basis method, so that the node displacement results of different design parameters are obtained by model mapping and low-degree matrix operation in the online calculation process. Compared with the traditional finite element method, this method avoids solving and iteration of large linear algebraic equations in the online stage, and the solving speed is improved by more than 50 times, and the error of the simulated deformation result is only 1% compared with the traditional finite element method, which can be used for fast and real-time simulation of specific equipment such as pressure vessels under various working conditions.

[0060] 3. The present application establishes an iterative solution format based on error analysis in offline solution. By trial calculation and error analysis on the test samples, the number of training samples of the reduced basis method and the Gaussian process regression model is gradually increased by iteration, so that the number of samples and the calculation time of offline calculation and the degree of freedom of online calculation can be minimized while ensuring the solution accuracy, and the overfitting phenomenon caused by too many training samples can be avoided. The calculation degree of freedom of the present application is the same as the number of training samples, and the numerical simulation results corresponding to different design parameters can be quickly and accurately simulated by simple calculation and mapping, and in addition to the simulation of elastic-plastic problems, it can also be used for fatigue, damage and other material nonlinear or geometric, contact nonlinear problems. BRIEF DESCRIPTION OF DRAWINGS

[0061] Figure 1 The technical flowchart of the present application.

[0062] Figure 2 The geometric model of Example 1 of the present application.

[0063] Figure 3 Meshing for Example 1 of the present application.

[0064] Figure 4 Iteration error curve for Example 1.

[0065] Figure 5 Displacement results for Example 1 using the method of the present application.

[0066] Figure 6 Displacement results for Example 1 using the finite element method.

[0067] Figure 7 Mises stress results for Example 1 using the method of the present application.

[0068] Figure 8 Mises stress results for Example 1 using the finite element method.

[0069] Figure 9 Equivalent plastic strain results for Example 1 using the method of the present application.

[0070] Figure 10 Equivalent plastic strain results for Example 1 using the finite element method.

[0071] Figure 11 Geometry model and meshing for Example 2 of the present application.

[0072] Figure 12 Displacement results for Example 2 of the present application for the first set of design parameters.

[0073] Figure 13 Displacement results for Example 2 of the present application for the first set of design parameters using the finite element method.

[0074] Figure 14 Mises stress results for Example 2 of the present application for the first set of design parameters.

[0075] Figure 15 Mises stress results for Example 2 of the present application for the first set of design parameters using the finite element method. DETAILED DESCRIPTION

[0076] The technical solutions and implementation processes of the present application will be described below in conjunction with the embodiments of the present application and the accompanying drawings. Obviously, the described embodiments are only used to explain the implementation processes and effects of the present application, and do not limit the application scope of the present application. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without creative efforts belong to the protection scope of the present application.

[0077] 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.

[0078] 1. Pressure vessel geometric model

[0079] 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.

[0080] 1. Design parameter definition

[0081] 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 A set of training samples is formed to avoid the efficiency loss of repeated modeling. Indicates the Design parameters, , Transpose flag for vectors.

[0082] 2. Establish the geometric model of pressure vessel

[0083] 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 .

[0084] 3. Finite element calculation

[0085] 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, represent corresponding node displacement results of each group training sample, .

[0086] Then, the corresponding node displacement results of each group training sample are stored in the current group training sample.

[0087] Finally, from all the training samples, a group is selected as the initial training sample, and from the remaining text, a group is selected as the test sample, and there are groups. .

[0088] II. Design parameter space

[0089] To solve the computational bottleneck of high-dimensional finite element equations, a reduced basis strategy is adopted, a reduced basis space is constructed through finite element simulation results, and orthogonal space basis vectors are obtained by Schmidt orthogonalization of the reduced basis space; the reduced basis solution is calculated through the orthogonal space basis vectors, the reduced basis solution space is constructed, and the design parameter space is constructed based on the training sample.

[0090] 1. Reduced basis space

[0091] 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 concentration is carried out in the form of reduced basis space. That is, a reduced basis space of dimension is constructed according to the selected initial training sample , represents the node displacement result corresponding to the th training sample, .

[0092] 2. Reduced basis solution

[0093] Schmidt orthogonalization is performed on the reduced basis space to avoid ill-conditioned matrix problems, thereby obtaining orthogonal space basis vectors , represents the dimension of the space, and the reduced basis solutions corresponding to the group initial training samples are calculated through the space basis vectors respectively as follows:

[0094] (1);

[0095] Where, the vector represents the reduced basis solution of .

[0096] 3. Design parameter space

[0097] Each reduced basis solution is combined in turn to form a reduced basis solution space , and the selected​ design parameter space corresponding to the initial training sample set , representing the first initial training sample set.

[0098] As a physical intelligent carrier, the key deformation modes are condensed from the finite element sample solution, and the orthogonal basis vector matrix is constructed by Schmidt orthogonalization to compress the million-dimensional freedom to the hundred-dimensional reduced order space. This not only filters numerical noise, but also directly calls the orthogonal space basis vector mapping (formula (9)) in the online prediction stage, which greatly reduces the grid model calculation time (such as 46 seconds to 1.64 seconds in Table 2), while strictly guaranteeing the physical reasonableness of the solution. Its adaptive evolution ability, that is, updating the orthogonal space basis vector when the sample increases, further supports real-time simulation in complex scenarios, and provides a bottom technology pillar for industrial real-time monitoring and optimization design.

[0099] Three, Gaussian process regression model

[0100] The Gaussian process regression method is used to establish the mapping relationship between the design parameter space and the reduced basis solution space, and the automatic correlation determination square exponential kernel function is selected as the kernel function of the Gaussian regression process. The maximum likelihood estimation is used to optimize the kernel function through parameter optimization, and the Gaussian process regression model representing the nonlinear mapping relationship between the reduced basis solution and the design parameter is established based on the optimized kernel function.

[0101] 1. Component modeling

[0102] In order to avoid overfitting caused by high-dimensional output, component modeling is adopted:

[0103] (2) ;

[0104] wherein, is the data set of the first component of the reduced basis solution, is the first component of , , represents the total number of components.

[0105] 2. Kernel function

[0106] The automatic correlation determination square exponential kernel function can adaptively quantify the differentiated influence of different design parameters (such as wall thickness and internal pressure) on the deformation of the pressure vessel, automatically identify the key sensitive parameters through the characteristic length parameter, and accurately fit the strong nonlinear relationship of the elastic-plastic deformation by virtue of the infinitely differentiable characteristic, so that the method can converge only by using hundreds of training samples under the premise of ensuring small displacement prediction error, realize millisecond online prediction, and perfectly solve the multiple contradictions between parameter sensitivity identification, nonlinear modeling and real-time performance of the traditional method. Therefore, the automatic correlation determination square exponential kernel function is used as the kernel function of the Gaussian regression method.

[0107] (3) ;

[0108] In the formula, is the characteristic length of the i-th design parameter, and each design parameter (such as wall thickness and internal pressure) has an independent characteristic length , so as to automatically quantify the influence weight of the design parameter on the deformation. is the standard deviation. is the prediction result of the i-th design parameter, and are the i-th design parameters of and respectively. is the matrix calculated by the kernel function on and . The covariance matrix is established as follows: (4) ;

[0109] (4) ;

[0110] In the formula, represents the covariance matrix; represents the matrix calculated by the kernel function on the design parameter space ; represents the noise variance; represents the unit matrix of the dimension. 3, maximum likelihood estimation optimization

[0111]

[0112] ​​​Maximum likelihood estimation can automatically balance model accuracy and generalization ability based on probability statistics principle: the method maximizes the likelihood function of the training samples, so that the Gaussian process regression model accurately captures the nonlinear relationship between design parameters and deformation response, while suppressing the numerical disturbance of finite element calculation through the noise term to avoid overfitting. Compared with traditional optimization methods such as grid search, maximum likelihood estimation can converge efficiently under hundreds of samples (such as 124 samples in Example 1), and the optimized kernel function parameters have clear physical meaning, providing quantitative basis for engineering decision-making. Therefore, by optimizing the related parameters of the kernel function through maximum likelihood estimation, the Gaussian process regression model is fitted , and the nonlinear mapping relationship between the reduced basis solution and the design parameter vector is established :

[0113] (5) ;

[0114] In the formula, represents the first component of the reduced basis solution.

[0115] Four, train the Gaussian process regression model through the finite element simulation results until the prediction accuracy requirement is met; and use the trained Gaussian process regression model to predict the finite element simulation results corresponding to the design parameters.

[0116] 1. Obtain the finite element simulation results using the Gaussian process regression model

[0117] First, select groups of training samples that are not used to fit the Gaussian process regression model from the remaining training samples as test samples , represents the group of test samples, .

[0118] The corresponding reduced basis prediction solution is calculated using the Gaussian process regression model:

[0119] (6) ;

[0120] In the formula, represents the first component of the reduced basis prediction solution of ; represents the reduced basis prediction solution of .

[0121] Then, map the reduced basis prediction solution back to the finite element space to obtain the predicted solution of the finite element simulation results, i.e. the node displacement results:

[0122] ​​​(7);

[0123] wherein, represents the predicted solution of the node displacement result obtained by mapping.

[0124] 2. Training the Gaussian process regression model

[0125] First, the relative error between the predicted solution of the node displacement result and the true value is calculated :

[0126] (8);

[0127] wherein, represents the second norm; represents the true value of the i-th test sample.

[0128] Then, the size relationship between the relative error and the set convergence tolerance is compared, if , the Gaussian process regression model converges, indicating that the Gaussian process regression model is trained; if , a new set of design parameters is constructed, and , steps 2 to 4 are executed in a loop until .

[0129] 3. Using the Gaussian process regression model to predict the finite element simulation result

[0130] For any new vector composed of a set of design parameters , the trained Gaussian process regression model is used to calculate the corresponding reduced basis solution, and the predicted solution of the corresponding node displacement result is obtained by mapping :

[0131] The calculation formula of the predicted solution is as follows:

[0132] (9);

[0133] wherein, represents the vector reduced basis solution obtained by the Gaussian process regression model.

[0134] Then, according to the basic theory of finite elements, the element strain matrix is calculated, the strain of the element or the Gaussian point is calculated , and is substituted into the corresponding constitutive calculation method to obtain the finite element simulation results of the pressure vessel, including stress and equivalent plastic strain , etc. ​

[0135] Through strict closed-loop verification and efficient online prediction mechanism, the precision and efficiency of pressure vessel deformation analysis are realized. First, based on the error feedback of test samples (formula (8)), the model is adaptively optimized, so that the displacement prediction error is reduced to 0.58% (example 1), and the online calculation complexity is compressed through orthogonal basis mapping (formula (9)), and the response speed of the ten-level grid model is greatly improved; second, the complete strain / stress field output (formula (10)) guarantees the engineering practicability, and the dynamic sample expansion strategy reduces 80% of the redundant calculation. Finally, on the premise of ensuring the accuracy of 99% of the elastic-plastic problem, the real-time simulation is realized in seconds (example 2 only needs 2.36 seconds), which provides an efficient tool for pressure vessel design optimization and safety monitoring. Figures 7 to 10

[0136] Five, examples:

[0137] 1. Example 1:

[0138] Fast prediction of elastic-plastic deformation behavior of cylinders of different sizes under internal and external pressure:

[0139] This example considers a typical structure of a pressure vessel, the elastic-plastic deformation behavior of a thick-walled cylinder under internal and external pressure, the geometric shape is shown in Figure 2 , the inner and outer surfaces are subjected to pressure and , respectively, the cylinder length and the relevant parameters of the bilinear elastic-plastic material are shown in Table 1.

[0140] Table 1 Model parameters and material properties of example 1

[0141] ;

[0142] In order to calculate the deformation and stress of cylinders of different sizes under different pressure loads, the method proposed in the present application is used for fast simulation. First, determine the design variables as , wherein the outer radius of the cylinder , the thickness of the cylinder , . The finite element model is shown in Figure 2 , the model is divided into 4000 elements and 5040 nodes, and the axial displacement of the end surface is constrained during simulation, as shown in Figure 3 .

[0143] Subsequently, the offline mode calculation of this example is carried out, the initial training sample number , the test sample number , the sample parameters composed of groups of design variables are given in the parameter range of the design variables, the finite element results are calculated, and 5 groups of samples are randomly selected to construct the reduced basis space​ , the Schmitt orthogonalization is implemented on the training samples to obtain , the reduced basis solution corresponding to each group of samples is calculated according to formula (1) , the mapping relationship between and is established by the Gaussian process regression method according to formulas (2)-(5).

[0144] Then, the node displacement results of the other 10 groups of test samples are predicted according to formulas (6)-(7) , and the errors are calculated according to formula (8) by referring to the corresponding finite element solutions. If the error is greater than , a group of sample parameters is added and the finite element solution is calculated, and the number of training samples is correspondingly increased , the reduced basis space is reconstructed until the error meets the convergence requirement.

[0145] Finally, the trained reduced basis space and the mapping relationship of the Gaussian process regression can be used for online prediction. For any parameter , its corresponding reduced basis solution can be obtained by mapping , and the node displacement result is calculated according to formula (9), and the stress and strain are calculated by substituting the constitutive relationship. The online prediction process does not need to solve large-scale linear algebraic equations when calculating the node displacement result, only matrix multiplication operation is needed, and then the strain and stress are calculated according to the node displacement result, which can greatly improve the solving efficiency, and since the finite element solution is used as the reference solution for model reduction in the offline stage, the accuracy of the result is guaranteed.

[0146] When the offline calculation of the present embodiment converges, the number of training samples used is 124 groups, and the relationship between the test error and the number of samples in the offline calculation process is shown in Figure 4 . Then, the trained pressure vessel geometric model is used to predict the node displacement results of , , and when the stress is 220 MPa and 110 MPa, respectively, and the strain and stress are calculated. In order to more intuitively give the stress and strain state of the material, the Mises equivalent stress and the equivalent plastic strain of the calculation unit are calculated as follows:

[0147] (10) ;

[0148] (11) ;

[0149] In the formula, the deviatoric stress tensor , is a second-order unit tensor, is the plastic strain tensor, is the Cauchy stress tensor. Figures 5 to 10 The structure displacement, Mises equivalent stress and equivalent plastic strain distribution results of the present application under the predicted parameters of the present embodiment calculated by the finite element method are given. Further, the node displacement error of the present application and the finite element solution is calculated according to formula (8) to be only 0.58%. In order to further illustrate the acceleration effect of the present application, the grid size is increased and decreased respectively, the model is trained, and the calculation time is shown in Table 2. It can be seen that the calculation time of the intelligent reduced basis finite element method proposed by the present application is basically unchanged, while the finite element method is greatly affected by the number of grids. This is because the time consumption of the online calculation of the present application is mainly used for the calculation of the unit strain and stress, and only matrix multiplication operation is needed when solving the displacement. While solving the linear algebraic equation set by the finite element method, the inverse of the stiffness matrix is needed, and the dimension of the reduced basis matrix is only related to the number of training samples r Regarding, And the stiffness matrix of the finite element is related to the displacement freedom. Therefore, the more the node freedom of the geometric model of the pressure vessel, the more obvious the acceleration effect of the present application compared with the traditional finite element method, which shows the effect of the fast calculation of the present application.

[0150] Table 2 Calculation time of the present application and the traditional FEM of example 1

[0151] ;

[0152] The present embodiment is aimed at the simulation analysis of the elastic-plastic mechanical behavior of the thick-walled cylinder of the pressure vessel under the conditions of internal pressure and external pressure, and the radius, thickness and internal pressure and external pressure of the thick-walled cylinder are taken as design variables. The accuracy and fast calculation effect of the present application are verified, which shows the potential of the present application in the fast simulation design of the pressure vessel.

[0153] 2. Example 2:

[0154] Fast prediction of elastic-plastic deformation behavior of horizontal pressure vessel under different loads:

[0155] The present embodiment considers the deformation of a horizontal pressure vessel structure under the parameters of internal pressure, gravity, seismic load, etc. The specific implementation manner is similar to that of example 1, which will not be described here. The model and mesh division of the present embodiment are shown in Figure 11 The model is divided into 65776 elements and 81546 nodes, of which 4 layers of grids are divided along the thickness direction of the tank body. In order to more clearly show the internal situation, only half of the model is shown. The materials of the saddle and the tank body are set as bilinear elastic-plastic materials, and the related mechanical properties are shown in Table 3.

[0156] Table 3 Material parameters of horizontal storage tank

[0157] ;

[0158] Table 4 gives the main geometric parameters of the storage tank structure.

[0159] Table 4 Main structure parameters of horizontal storage tank

[0160] ;

[0161] In the simulation, the bottom end of the storage tank saddle is fixed, and the load conditions that need to be considered in the design of the tank body are referred to, the tank body is considered to be subjected to internal uniform pressure , the whole is subjected to gravity and three-directional seismic acceleration , and , the design parameters , wherein , , and m / s 2 , , the size is 9.8m / s 2 , the direction is the negative direction of the axis.

[0162] The embodiment first carries out offline mode calculation, defines that the initial training sample number is 5 and the test sample number is 10, when the training sample , the prediction model converges, the average node displacement error obtained by calculating the test set is 0.4%. The model after convergence is used to calculate three groups of different design parameters, and the design parameters, simulation errors and calculation time are as shown in Table 5.

[0163] Table 5 Comparison of main structure parameters and calculation results of horizontal storage tank

[0164] ;

[0165] Further, the stress and displacement results obtained by the present application and the finite element method under three conditions are given, and the results of the first group of design parameters are as shown in Figures 12 to 15 , and similar to the previous, only half of the model is displayed in order to more clearly show the displacement and stress conditions inside the model. It can be seen in combination with Figure 14 , Figure 15 and material parameters that for the first group of design parameters, the tank body is in an elastic stage and does not occur plastic deformation, and for the last two groups of load parameters with higher internal pressure, the tank body occurs plastic deformation in different degrees. The results of the present application and the finite element method show that the present application can quickly and accurately predict the stress and deformation conditions of the tank body for the three conditions.

[0166] In the embodiment, the model obtained through offline calculation can be used to obtain simulation results quickly through online calculation for each different design parameter, the time cost is only 1% of that of the finite element method, and the error is only 1%, which proves that the application can quickly and accurately simulate the elastic-plastic deformation behavior of the pressure vessel under the action of loads such as earthquakes and internal pressure.

[0167] Through the simulation of the two embodiments, it is shown that the application has excellent performance in quickly simulating the elastic-plastic deformation of the pressure vessel, and it is proved that the application has great potential in real-time and quick simulation of the pressure vessel. The application can quickly obtain simulation results of different geometric and load parameters in the design process of the pressure vessel, and increase the efficiency of the design of the pressure vessel, and can also perform real-time and quick simulation of the deformation and stress state of the pressure vessel based on the real-time monitored load data.

[0168] The above is only the preferred specific embodiment of the application, but the protection scope of the application is not limited to this, any person skilled in the art can make equivalent replacement or change according to the technical scheme and the inventive concept of the application within the technical range disclosed by the application, which should be covered in the protection scope of the application.

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 the mapping relationship between the design parameter space and the reduced basis solution space. At the same time, select the automatic correlation determined square exponential kernel function 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. The automatic correlation determined square exponential kernel function is expressed as follows: Where, is the standard deviation, For the The characteristic length of the design parameter, Indicated by A vector of design parameters; for The prediction results, and They are and No. Design parameters; Indicates that the kernel function is and The matrix calculated above; 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; 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 Group the training samples that are not used to fit the Gaussian process regression model 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.

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