A response prediction method for a complex structure fine finite element model

Abstract

The application provides a response prediction method of a complex structure fine finite element model, comprising the following steps: establishing a fine finite element model and a rough mirror information model with different grid densities, which represent the system characteristics of the physical model of the structure; using a Latin hypercube sampling method for random sampling to construct input parameter sample sets with sizes of m and n respectively; performing a probability finite element analysis to extract corresponding output response sample sets; constructing a Kriging model with the first m groups of data in the output response sample sets, and using a verification error to evaluate the prediction accuracy; and predicting the output response of the fine finite element model corresponding to the remaining n-m groups of data in the rough mirror information model response sample set according to the Kriging model. The application effectively reduces the dependence of the efficiency of the proxy modeling on the fineness of the forward calculation model, and greatly reduces the calculation time of the complex structure fine finite element model for obtaining system response.

Description

Technical Field

[0001] The present invention relates to the technical field of structural engineering, and in particular to a method for predicting the response of a fine finite element model of a complex structure by using a rough mirror image information model driven by a Kriging model. Background Art

[0002] In the real world, the preliminary safety and stability assessments of many complex structures rely on numerical simulations. One of the most commonly used methods is to create detailed finite element models of complex structures to help engineers study the actual behavior and performance of the structures as closely as possible.

[0003] However, there are also many disadvantages in using detailed structural finite element models for numerical simulation. For example, the efficiency of numerical calculations is heavily dependent on the refinement of the structural finite element model, which causes the time to obtain the response of the detailed structural finite element model to increase dramatically with the complexity of the structural finite element model. Especially when it is necessary to perform dynamic response analysis (modal analysis, transient analysis, etc.), the use of detailed structural finite element models undoubtedly places extremely stringent requirements on the computer's memory and computing power.

[0004] To this end, this application proposes a method for predicting the response of a fine finite element model of a complex structure using a rough mirror image information model. Summary of the Invention

[0005] To solve the above problems, the present invention proposes a method for predicting the response of a complex structure fine finite element model based on a rough mirror information model driven by a Kriging model.

[0006] To achieve the above objectives, the present invention provides the following technical solutions.

[0007] A response prediction method for a complex structure fine finite element model comprises the following steps:

[0008] According to the complex structure to be predicted, a fine finite element model is established to characterize the system characteristics of the structural physical model, and a coarse mirror information model that is twinned with the fine finite element model and has different mesh density is established;

[0009] Based on prior knowledge, the probability distribution types obeyed by the material parameters in the fine finite element model and the rough mirror information model are determined, and Latin hypercube sampling is used for random sampling to construct input parameter sample sets with sizes m and n respectively.

[0010] According to the input parameter sample set, probabilistic finite element analysis is performed on the rough mirror information model and the fine finite element model respectively, and the corresponding output response sample set is extracted;

[0011] The Kriging model is constructed based on the first m groups of data in the output response sample set of the rough mirror information model and the fine finite element model, and the prediction accuracy is evaluated using the verification error to reconstruct the Kriging model;

[0012] The Kriging model is used to predict the output response of the fine finite element model corresponding to the remaining nm group data in the rough mirror information model response sample set.

[0013] Preferably, the input parameters of the input parameter sample set include material parameters and boundary conditions; wherein n≥5m.

[0014] Preferably, the output response includes frequency, displacement and stress.

[0015] Preferably, the Kriging model As shown in the following formula:

[0016]

[0017] Where θ T is the transpose of the corresponding regression coefficient vector, F(x)=[F1(x),...,F M (x)] is the polynomial basis function, θ T F(x) is the trend of the Kriging model, and G(x) is a Gaussian process with zero mean.

[0018] Preferably, the construction of the Kriging model further includes the following steps:

[0019] Construct the covariance function of G(x) to associate it with the hyperparameters in the Kriging model;

[0020] Calibrate the Kriging model The hyperparameters in .

[0021] Preferably, the construction of the covariance function of G(x) comprises the following steps:

[0022] Define G(x):

[0023] G(x)=Cov(G(x i ), G(x j ))=σ 2 R(x i , x j ;θ)

[0024] Where x i and x j is a pair of sampling points in the structure output response sample space, G(x i ) and G(x j ) are the observed value and the new interpolation value respectively; σ2 is the constant variance of G(x); R(x i , x j ; θ) is the correlation function, describing G(x i ) and G(x j ) and the correlation coefficient θ=[θ1,...,θ n ] T similarities between;

[0025] Matérn-5 / 2 is used as the correlation function, and its formula is as follows:

[0026]

[0027] Preferably, the calibration Kriging model The hyperparameters in the , including the following steps:

[0028] consider Assuming that it obeys a multivariate Gaussian distribution, the unknown hyperparameters γ in the Kriging model are estimated by maximizing the likelihood function. 2 ,θ), as shown below:

[0029]

[0030] Where C = σ 2 R+∑ n is the covariance matrix, ∑ n is the noise response, P=[p(x1),...p(x N )] T is element P ij =p j (x i )’s N×M regression matrix;

[0031] For the above formula about θ and σ 2 Find the partial derivative and set it to zero, and the solution for θ is transformed into solving the following optimization problem:

[0032]

[0033] Where D θ is the parameter space of θ, R is R(x i , x j ; abbreviation of θ).

[0034] Preferably, the method of evaluating the prediction accuracy by using the validation error and reconstructing the Kriging model comprises the following steps:

[0035] The leave-one-out cross-validation error is used to evaluate the accuracy of the Kriging model until the model accuracy meets the requirements; otherwise, the input-output data set of the structure to be analyzed is repeatedly constructed, and the Kriging model is rebuilt until the model accuracy meets the preset requirements;

[0036] Among them, the accuracy of the Kriging model is evaluated by the leave-one-out cross-validation error according to the following formula:

[0037]

[0038] Where, is the jth th The structure output response sample value at the sample point, Is to exclude the jth th The structural output response prediction value of the Kriging model at the sample point is is the mean of the sample set of structure output responses.

[0039] Beneficial effects of the present invention:

[0040] The method provided by this invention uses a Kriging model-driven, coarse mirror image information model to predict the response of a fine finite element model of a complex structure. This method takes into account the a priori uncertainty of input parameters and reduces the assumed errors in the finite element model parameters by performing probabilistic finite element analysis. By establishing a coarse mirror image information model corresponding to the fine finite element model of a complex structure and utilizing generative AI technology to expand the proxy modeling database, the method effectively reduces the dependency of the proxy modeling efficiency on the refinement of the forward computational model, significantly reducing the computational time required to obtain the system response from the fine finite element model of a complex structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 This is a flow chart of a method for predicting the response of a complex structure fine finite element model using a rough mirror image information model driven by a Kriging model according to an embodiment of the present invention;

[0042] Figure 2 A schematic diagram of the basic principle of an embodiment of the present invention;

[0043] Figure 3 The fine finite element model M-1 and its twin rough mirror image information model M-2 of the embodiment of the present invention;

[0044] Figure 4 This is a layout diagram of important sampling points on the dam body according to an embodiment of the present invention;

[0045] Figure 5 Graph showing the Kriging model verification and prediction of the X-direction displacement responses of M-1 and M-2 at sampling point 1 according to an embodiment of the present invention;

[0046] Figure 6 This is a scatter plot of the true response of the first 20 natural frequencies of M-1 and the Kriging model response in Example 1 of the present invention. DETAILED DESCRIPTION

[0047] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0048] Example 1

[0049] The present invention provides a response prediction method for a complex structure fine finite element model, the specific steps are as follows: Figure 1 As shown, the following steps are included:

[0050] S1: A complex structure with high computational cost is selected as the research object, and two sets of finite element models with different mesh densities are established for the research object, which are respectively denoted as M-1 and M-2. M-1 is a fine finite element model that can characterize the system characteristics of the structural physical model, and M-2 is a rough mirror information model of M-1. The basic principle diagram is shown as follows: Figure 2 shown.

[0051] S2: Based on prior knowledge, the probability distribution type obeyed by the material parameters in M-1 and M-2 is determined, and then the Latin hypercube sampling method is used to perform random sampling in its probability space to construct input parameter sample sets of sizes m and n for M-1 and M-2 respectively (n≥5m).

[0052] S3: Perform probabilistic finite element analysis and extract the corresponding output response sample sets in M-2 and M-1, which are recorded as X and Y respectively.

[0053] S4: Construct a Kriging model based on the first m groups of data in X and Y, and use the validation error to evaluate its prediction accuracy.

[0054] S5: Predict Y corresponding to the remaining nm groups of data in X based on the high-precision Kriging model.

[0055] Among them, the Kriging agent model is established The steps include:

[0056] The Kriging surrogate model is constructed using the first m groups of output response sample data in M-1 and M-2; the leave-one-out cross-validation error is used to evaluate the accuracy of the Kriging model until the model accuracy meets the requirements (generally speaking, in pure numerical problems, Err LOO ≤1.0E-5, Err in engineering problemsLOO ≤1.0E-3); otherwise, the “input-output” data set of the structure to be analyzed is repeatedly constructed, and then the Kriging proxy model is re-established until the model accuracy meets the requirements.

[0057] The Kriging surrogate model is constructed using the following formula

[0058]

[0059] Where θ T represents the transpose of the corresponding regression coefficient vector, F(x) = [F1(x), ..., F M (x)] is a polynomial basis function, θ T F(x) is the trend of the Kriging model, and G(x) is a Gaussian process with zero mean.

[0060] Furthermore, the Kriging surrogate model The steps also include:

[0061] Construct the covariance function of G(x) to associate it with the hyperparameters in the Kriging model;

[0062] Calibrate the Kriging model The hyperparameters in .

[0063] Specifically, constructing the covariance function of G(x) includes:

[0064] Define G(x) as follows:

[0065] G(x)=Cov(G(x i ), G(x j ))=σ 2 R(x i , x j ;θ)

[0066] Where x i and x j is a pair of sampling points in the structure output response sample space, G(x i ) and G(x j ) are the observed value and the new interpolation, σ 2 is the constant variance of G(x), R(x i , x j ; θ) is the correlation function, which describes G(x i ) and G(x j ) and the correlation coefficient θ=[θ1,...,θ n ] T The "similarity" between them.

[0067] Select a suitable correlation function. In this study, the Matérn-5 / 2 correlation function with strong smoothness and high versatility is selected. Its formula is as follows:

[0068]

[0069] Specifically, calibrate the Kriging model The hyperparameter steps in include:

[0070] consider Assuming that it obeys a multivariate Gaussian distribution, the unknown hyperparameters γ in the Kriging model are estimated by maximizing the likelihood function. 2 ,θ), as shown below:

[0071]

[0072] Where C = σ 2 R+∑ n is the covariance matrix, where ∑ n is the noise response, P=[p(x1),...p(x N )] T is element P ij =p j (x i ) is an N×M regression matrix.

[0073] For the above formula about θ and σ 2 Find the partial derivative and set it to zero, so the solution for θ can be transformed into solving the following optimization problem:

[0074]

[0075] Where D θ is the parameter space of θ, R is R(x i , x j ; abbreviation of θ).

[0076] Specifically, the accuracy of the Kriging model is evaluated using the leave-one-out cross-validation error according to the following formula:

[0077]

[0078] In the formula is the jth th The structure output response sample value at the sample point, Is to exclude the jth th The structural output response prediction value of the Kriging model at the sample point is is the mean of the sample set of structure output responses.

[0079] In this embodiment:

[0080] This example uses a real, large, and complex high arch dam structure as an example to investigate the prediction of the fine finite element model response of a complex structure using a coarse mirror information model driven by a Kriging model. The high arch dam is a 95-meter-high concrete hyperbolic arch dam with three spillway openings located in the middle of the dam crest. The crest elevation is 643.5 meters, and each opening has a clear width of 10 meters. Construction of the dam was completed in November 2015, and water impoundment began in February 2017, completing at the end of July 2017.

[0081] The fine finite element model M-1 of the arch dam and its twin rough mirror image information model M-2 were both constructed using hexahedron and tetrahedron SOLID185 elements in the large commercial modeling software HyperMesh. To simplify the complexity of the problem, the high arch dam structure only considers two major partitions: the dam body and the foundation. The specific drawings are shown in Figure 3 , M-1 and M-2 have the same characteristics (loads, boundary conditions, material parameters, etc.) except for the different mesh densities. Table 1 gives the basic mesh parameters of the two.

[0082] Table 1 Basic grid parameters of high arch dams M-1 and M-2

[0083] M-1 M-2 <![CDATA[N M-2 / N M-1 ]]> Number of dam units 42712 675 1.58％ Total number of units 201728 2171 1.08％

[0084] In this embodiment, the superiority of the method shown in the present invention is verified mainly from two aspects: structural static analysis and modal analysis. First, a probability input prior model is set for the material parameters in the model based on prior knowledge, as shown in Table 2; then, 500 groups of probability sampling of material parameters are performed in this parameter probability space using the Latin hypercube sampling method, where M-1 uses the first 100 groups to perform probabilistic finite element analysis, and M-2 uses the full-scale material parameter sample set to perform probabilistic finite element analysis; then, the corresponding output response sample sets are extracted, that is, M-1 corresponds to 100 groups of sample data and M-2 corresponds to 500 groups of sample data; a Kriging model is constructed based on the first 100 groups of output response samples in M-1 and M-2, where the first 50 groups are used for experimental design and the last 50 groups are used to verify the accuracy of the Kriging model; finally, the generated high-precision Kriging model is used to predict the output response of M-1 based on the remaining 400 groups of output responses in M-2.

[0085] In this embodiment, all numerical calculation environments are based on a high-performance UNIX workstation with dual nodes, each node has a 36-core CPU and 192 GB memory. The calculation software uses ANSYS APDL, which executes finite element analysis by calling 12 threads. The output responses extracted from the structural static analysis are M-1 and M-2 corresponding to Figure 4The X-displacement at sampling point 1 in the figure. The output responses extracted by modal analysis are the first 20 natural frequencies of M-1 and M-2. Figure 5 This is the Kriging model verification and prediction diagram of the X-direction displacement response of M-1 and M-2 at sampling point 1. It can be clearly seen from the figure that the 50 groups of verification sets and the corresponding Kriging model prediction sets completely overlap. The Kriging model leave-one-out method verification error Err corresponding to the displacement response is LOO =1.08E-08, which fully meets the accuracy requirements. Figure 6 The scatter plot of the first 20 natural frequency real response verification sets of M-1 and the Kriging model response prediction sets shows that both are basically distributed on the scatter trend line, and R 2 The ultra-high precision of the Kriging model in these two aspects demonstrates the effectiveness and excellent performance of the method proposed in this invention.

[0086] Table 2 Probabilistic input prior model of material parameters of high arch dam M-1 and M-2

[0087]

[0088] Table 3 Calculation time for obtaining output responses of high arch dams M-1 and M-2

[0089]

[0090] The above embodiments demonstrate that the present invention effectively reduces the dependence of the efficiency of proxy modeling on the precision of the forward computation model, and greatly reduces the computation time for obtaining system responses for complex structures.

[0091] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A response prediction method for a complex structure fine finite element model, characterized in that: The following steps are involved: According to the complex structure to be predicted, a fine finite element model is established to characterize the system characteristics of the structural physical model, and a coarse mirror information model that is twinned with the fine finite element model and has different mesh density is established; Based on prior knowledge, the probability distribution types obeyed by the material parameters in the fine finite element model and the rough mirror information model are determined, and Latin hypercube sampling is used for random sampling to construct input parameter sample sets with sizes m and n respectively. According to the input parameter sample set, probabilistic finite element analysis is performed on the rough mirror information model and the fine finite element model respectively, and the corresponding output response sample set is extracted; The Kriging model is constructed based on the first m groups of data in the output response sample set of the rough mirror information model and the fine finite element model, and the prediction accuracy is evaluated using the verification error to reconstruct the Kriging model; The Kriging model is used to predict the output response of the fine finite element model corresponding to the remaining nm group data in the rough mirror information model response sample set; The Kriging model As shown in the following formula: Where, is the transpose of the corresponding regression coefficient vector, F(x)=[F1(x),…,F M (x)] are polynomial basis functions, is the trend of the Kriging model, G(x) is a Gaussian process with zero mean; The construction of the Kriging model further includes the following steps: Construct the covariance function of G(x) to associate it with the hyperparameters in the Kriging model; Calibrate the Kriging model Hyperparameters in ; The construction of the covariance function of G(x) comprises the following steps: Define G(x): G(x)=Cov(G(x) i ),G(x j ))=σ 2 R(x i ,x j ;θ) Where x i and x j is a pair of sampling points in the structure output response sample space, G(x i ) and G(x j ) are the observed value and the new interpolation value respectively; σ 2 is the constant variance of G(x); R(x i ,x j ; θ) is the correlation function, describing G(x j ) and G(x j ) and the correlation coefficient θ=[θ1,…,θ n ] T similarities between; The correlation function is the Matérn-5 / 2 correlation function, as shown below: The calibration Kriging model The hyperparameters in the , including the following steps: consider Assuming a multivariate Gaussian distribution, the unknown hyperparameters in the Kriging model are estimated by maximizing the likelihood function. As shown in the following formula: Where C = σ 2 R+∑ n is the covariance matrix, ∑ n is the noise response, P=[p(x1),...p(x N )] T is element P ij =p j (x i )’s N×M regression matrix; About the above formula and σ 2 Find the partial derivative and set it to zero, and the solution for θ is transformed into solving the following optimization problem: Where D θ is the parameter space of θ, R is R(x i ,x j ; abbreviation of θ).

2. The response prediction method of a complex structure fine finite element model according to claim 1, characterized in that: The input parameters of the input parameter sample set include material parameters and boundary conditions; wherein n≥5m.

3. The response prediction method of a complex structure fine finite element model according to claim 1, characterized in that: The output responses include frequency, displacement, and stress.

4. The response prediction method of a complex structure fine finite element model according to claim 1, characterized in that: The method of using validation error to evaluate prediction accuracy and reconstructing the Kriging model includes the following steps: The leave-one-out cross-validation error is used to evaluate the accuracy of the Kriging model until the model accuracy meets the requirements; otherwise, the input-output data set of the structure to be analyzed is repeatedly constructed, and the Kriging model is rebuilt until the model accuracy meets the preset requirements; Among them, the accuracy of the Kriging model is evaluated by the leave-one-out cross-validation error according to the following formula: Where, is the jth th The structure output response sample value at the sample point, Is to exclude the jth th The structural output response prediction value of the Kriging model at the sample point is is the mean of the sample set of structure output responses.

Images