Performance Comparison of Two Meta-Model for the Application to Finite Element Model Updating of Structures

Yang Liu，De-Jun Wang，Jun Ma，Yang Li

(School of Transportation Science and Engineering，Harbin Institute of Technology，Harbin 150090，China)

1 Introduction

FE model updating of structures has received considerable attentions in recent years due to its crucial role in fields ranging from establishing a realityconsistent structural model for dynamic analysis and control，to provide an accurate baseline model for damage detection and localization，etc.Structural FE model updating is to produce a refined FE model that could better predict the dynamic or static behavior of structures by minimizing the difference between the test data and analytical results obtained by using the analytical FE model.Therefore，F(xiàn)E model updating of structures usually ends up with an optimization problem［1－2］. Since the relationship between the characteristics of structures and structural parameters is usually nonlinear，it is inevitable to use a large number of iteration for solving the optimization issue of FE model updating.Furthermore，the more complicated the structure is，the larger numbers of iteration the algorithm needs.Due to above reasons，it is necessary and essential to improve the efficiency of updating the FE model of large-scale and complex structures.

For the real complex structures，the low efficiency of model updating mainly lie in the large number of degree of freedom(DOF)and the huge size of FE model.Therefore，itiseffective to improve the efficiency of FE model updating by the‘indirect usage’ of the analytical FE model.Herein，the‘indirect usage’means to generate a surrogate model(metamodel)of the analytical FE model of structures，which is the relationship between the static ofdynamic characteristics of structural and the structural parameters. The meta-model usually takes a mathematical function with some coefficients，obtained by regression method，to be the approximation of above functional relationship，and then the dynamic analysis of the analytical FEM could be replaced by this meta-model.

There are mainly three types of meta-model，i.e.，artificial neural network(ANN)［3］，Kriging model［4］and RSM［5］，and these meta-model have been applied to many different fields successfully［6－7］.To replace the FE model of structures，the ANN is apt to build a metamodel with a large number of structural parameters，and the generation of ANN always needs a large number of training samples.However，in order to make an over-determined optimization problem of FE model updating，the number of updating parameter is limited since it has to be smaller than the one of measured features.According to above reasons，the ANN is rarely applied to build the meta-model for the sake of FE model updating of structures.How are the Kriging model and the RSM?To investigate the application of Kriging model and RSM to FE model updating of structures，the performance of Kriging model and RSM are compared in detail.Some key issues of the application of meta-model to FE model updating of structures are proposed.We found that all the samples obtained by FE model for building the meta-model have no random error，so some methods based on statistical theory could not be applied to assess the accuracy of metamodel，especially for RSM.Finally，some advices are presented in order to select a reasonable meta-model for FEM updating of structures.

To this end，the remainder of this paper is organized as follows.The basic idea of FE model updating is introduced in next section.The detail of RSM and Kriging model are described in Section 3.Some key issues of the application of meta-model to FE model updating are discussed in Section 4，and then the FE model of a truss bridge model is updated by using the procedure of model updating based on metamodel.At last，conclusions are drawn.

2 Finite Element Model Updating

FE model updating of structures aims to correct the modeling error of the analytical FE model by minimizing the difference between the measured static or dynamic data and the results obtained by the analytical FE model.The difference between the measured and analytical results is defined as the following equation.

whereWεis the weighing matrix，and the usual form is defined as，

where Ζmrepresents the measured results such as the modal parameters，and［diag(·)］and diag(·) represent the diagonal matrix and the principal diagonal elements of matrix respectively.The ε in Eq.(1) means the difference between the measured results and the analytical results，which is defined as，

where Ζ(θ)are the analyticalstatic or dynamic characteristics of structures，and θ are the updating parameters for FE model updating.With the first-order Taylor series，the approximation to Eq.(3)is obtained by，

where θiis the updating parameter at the ith iteration，G(θi)is the derivative of Ζ(θ)with respect to θi.

In this study，we investigate to build a surrogate model of the analytical FE model，so the characteristics of structures Ζ(θ)shown in Eq.(3)is obtained by this surrogate model(meta-model)instead of the FE model analysis.With this procedure，the efficiency of FE model updating could be improved dramatically.

3 Frame of Meta-Model

The basic idea of meta-model is to build a surrogate model by using regression technique with limited feature samples，and different mathematical form goes to different meta-model.The procedure of building the meta-model consists of three parts，i.e.，i)selection of thefeaturesamples in structural parameters space， ii)fitting the mathematical functional model by using the selected feature samples，iii)assessment of the generated meat-model.

3.1 Selection of Feature Samples

Selection of the features samples is also called design of experiments(DOE).Using the selected samples of structural parameters，the response function (the static or dynamic characteristics of structures)is obtained by substituting the samples into the analytical FE model，and then the response function and the samples of structural parameters are applied to build the meta-model. Therefore， the proper selected samples are essentialforthe efficiency and the accuracy of generated meta-model.Couples of methods have been applied to select the feature samples such as full factorial design［8］，fractional factorial design［9］，central composite design［4］，orthogonal array［10］etc.Among these methods， the Latin hypercube sampling［11］is a popular strategy for generating random sample points ensuring that all portions of the sample space is represented.Herein，this method is applied and thedetail of theprocedureis described in Appendix A.

3.2 Mathematical Form of Meta-Model

Since RSM and Kriging model are two main metamodel for FE model updating of structures，the detailed mathematical form of these two models are discussed as follows.Here，we assume that the set of feature samples of updating parameters is Γ=(［θ1，θ2，…，θN］Τ)N×q，and each vector of updating parameters is θv={θv1，θv2，…，θvq}Τ(v=1，2，…，N)(N is the number of total samples，and q is the number of updating parameters).The corresponding response function obtained by the analytical FE model is Ζj(Γ) ={Zj(θ1)，Zj(θ2)，…，Zj(θN)}Τ(j=1，2，…，p，and j represents the number of measured modal parameters or measured displacement etc.).

3.2.1 Response surface model

The RSM is a common meta-model since it is easy to implement.For example，if we take the modal parameters as the response function，the RSM is defined as，

where ψ is the response function.The second-order polynomial is the popular RSM，e.g.，the structural parameter vector is θv={θv1，θv2}，and then the following equation is obtained，

where εLis the error term，{β0，β1，β2，β11，β22，β12} represents the coefficients of RSMθv5=θv1θv2.Above equation could be replaced by，

where，

The regression technique could be applied to obtain the coefficients of RSM，and then the RSM is generated.Utilizing the generated RSM，the estimated set of characteristics of structures is obtained as，

3.2.2 Kriging model

With the normalization method，the mean and variance of the selected feature samples are defined as，

where E(·)and V(·)represent the mean and variance respectively.

With the Kriging model，the estimation of the static or dynamic characteristic of structures Zj(θk)are divided into two parts，i.e.，the trend term and the random term，which is defined as［12－13］，

where fj(θv)usually is a polynomial.The random item γj(θv)is a random process，and its mean is defined as，

And the covariance is defined as，

where w∈Γ，and Cor(·)is the correlation function with respect to ρj={ρj1，ρj2，…ρjq}Τ.If w?Γ，above covariance is defined as，

The correlation function shown in above two equations has no unique form，and the common correlation function is the Gaussian function，i.e.，

With above equations，the coefficients of the Kriging model could be estimated by optimization method，and the detail of the estimation of these coefficients are described in Appendix B.

3.3 Assessment of Meta-Model

To assess the accuracy of the generated metamodel，a common way is to compare the error between the true values and estimated response function at extra testsamples. The index SST (Totalsum of squares)［14］is always used，and SST is composed of two parts，i.e.，SSR(Sum square of regression)and SSE(Sum square of error)，i.e.，

All the detail of above three indexes and the procedure of assessing the meta-model［15－16］are described in Appendix C.

4 Key Issues of the Application of Meta-Model to FE Model Updating of Structures

4.1 Statistical Meaning of the Generated Meta-Model

Since the RSM is based on the statistical theory，the error term in Eq.(11)consists two parts，i.e.，modeling error and random error respectively，

where εL1represents the modeling error causing from the difference between the RSM and the true model，and εL2represents the random error causing from some uncertainty such as the effect of noise or unavoidable artificial factors.

However， for the FE model updating of structures，all the estimated response variableshown in Eq.(18)are obtained by the deterministic FE model analysis.Therefore，there is no random error εL2at all，and only the modeling error εL1exist(Eq.(19))which would be zero if the mathematical form of RSM could take infinite order polynomial or this mathematical form is exactly same with the true model.

As shown in above equation，in theory，the estimated response variableis equal to the true value if the modeling error εL1goes to zero，and the estimated response variablehas the unique value for each feature sample.Therefore，the response variable)is not the random variable，as shown in Fig.1.

Following this logic，if the RSM is applied to replace the analytical FE model during model updating，all the methods introduced in Section 3.3 cannot be worked any more since the generated RSM in this situation have no statistical meaning.

Above issue is easy to be ignored when the RSM is applied to update the FE model of structures.Since all the methods based on statistical theory do not work at all，the only way to assess the accuracy of the predictor of RSM under the given polynomial form is to check the difference between the estimated response variable and the true value.Meanwhile，the methods obtaining the coefficients of RSM are not the regression technique but the optimization methods such as least square method.

Fig.1 Regression analysis without random error

Different with the RSM，the Kriging model does not lack the random error.As shown in Eq.(12)，the key difference between RSM and Kriging model lies in the random item γj(θv)，and the Kriging model consist of two parts(Fig.2)，i.e.，trend term fj(θk)and random term γj(θv).Based on the trend term shown in Fig.1，the random term is in charge of the accuracy of the Kriging predictor.The item γj(θv)is arbitrary function satisfying the equations from Eq.(14)to Eq.(16)，which means that γj(θv)has no deterministic mathematical form.And γj(θv)usually is a stochastic process.Therefore，the approaches based on statistical theory work well for assessing and determining the mathematical form of Kriging model.

Fig.2 Sketch map of the idea of the Kriging model

4.2 Selection of the Meta-Model for FE Model Updating of Structures

In theory，the selection the meta-model depends on the relationship between structural characteristics and structural parameters.For the issue of updating the FE model of structures，the structural parameters are the updating parameters that representing the modeling error between the analyticalFE modeland real structure.Using theTaylorseries，the structural characteristics is describe as the polynomial with respect to updating parameter，i.e.，

whereθ0is the initialupdating parameters，G represents the derivative matrix that is defined as follows.

where j=1，2，…，p represents the number of the measured structuralcharacteristicssuch asmodal parameters，and τ =1，2，…，q is the number of updating parameters.For the dynamic characteristics of structures，thederivativeof the eigen-value with respect to parameter θ is defined as［17］，

where λjis the jth eigen-value of structures，M and K representthe mass matrix and stiffness matrix respectively.The derivative ofeigen-vector with respect to parameter θ is defined as，

where φjis the jth eigen-vector of structures，Nu is the number of DOF of structures，and α is the modal participant coefficients which is defined as，

As shown in above equations，the nonlinear degree of the functional relationship shown in Eq.(20)changes with the variation of the updating parameters of structures.If the updating parameter is the coefficient of the element stiffness matrix，the second-order derivative shown in Eq.(20)goes to zero，and then the relationship between the response function and the updating parameter is approximate to linear function.If the length of the element l is taken as the updating parameter，the response function shown in Eq.(11)is the cubic function with respect to l. Furthermore，as we know，the larger the amounts of updating parameters increase，the higher the nonlinear degree of Eq.(20)becomes.

On the other hand，the nonlinear degree of the relationship shown in Eq.(20)also depends on the characteristics of structures.For example，there is the following relationship between the eigen-value of structures and vibration frequency of structures，

where ωjis the jth mode of structural frequency.According to above equation，comparing with λj，the nonlinear degree between ωjand θ is more serious.

In summary，to select a reasonable meta-model for the purpose of FE model updating，it is significant to considerthe nonlineardegree ofthe relationship between the characteristicsofstructuresand the updating parameters.The nonlinear degree of above relationship depends on the selected updating parameters as well as the structural characteristics.Therefore， a reasonable meta-model should be determined case by case，and the complication of the structures，the selected updating parameters and the goal of FE model updating(different characteristics of structures)are all the key factors for selecting the meta-model.

4.3 Advices of the Selection of the Meta-Model for FE Model Updating of Structures

From the discussion shown above，the following advices about the selection of the meta-model for FE model updating are drawn.

(1)The selection of the meta-model depends on the nonlinear degree of the relationship between the static or dynamic characteristics of structures and the updating parameters of structures，i.e.，the selected updating parameters，the goal of FE model updating (structural characteristics)and the complexity of the structures determine what type of the meta-model should be utilized for FE model updating.

(2)If it is difficult to utilize extra test sample for assessing the generated meta-model，leave-h-out crossvalidation method is an effective choice.

(3)If the RSM is applied to FE model updating of structures，all the approached described in Section 3.3 and Section 3.4 cannot be used at all.The only way to evaluate the accuracy of the generated RSM is to check the error between the predicted response function and the true values with respect to the test samples since the regular assessment method based on statistical theory have no statistical meaning any more.

(4)To the large-scale or complicated structures， e.g.long-span bridges，the relationship between the characteristics of structures and structural parameters always is a high nonlinear function，so it is recommend to utilize the Kriging model，which has no issue of lacking the statistical meaning，to replace the FE model of structures.

4.4 Procedure of the FE Model Updating Based on Meta-Model

The procedure of FE model updating based on meta-model is described as follows.

(1)Determine the set of feature samples Γ by using the methods introduced in Section 3.1.

(2)Obtain the set of characteristics of structures with respect to the selected feature samples Ζ(Γ)by using the analytical FE model of structures.

(3)Generate the meta-model by using the Γ and Ζ(Γ).

(4)Determine the final meta-model and assess the confidence level of the generated meta-model by using the methods shown in Section 3.3.

(5)Replace the analytical FE model with the generated meta-model and Update the FE model of structures by using the methods described in Section 2.

5 Experimental Example

In this section，we do not select a complex structure as an example.To describe the procedure of FE model updating based on meta-model，the FE model of a truss bridge model is updated by using the measured modal parameters.

5.1 Introduction of a Truss Bridge Model

A truss bridge model was built in the Smart Structures Technology Laboratory in the University of Illinois at Urbana-Champaign.This is a bolt-jointed steel bridge model with 56 notes and 164 members.The structure has 14 bays with each bay of 0.4 m long.Section of the frame is 0.4 m width by 0.4 m height.The member is steel tube with section of 1.71 mm in outer diameter，and 3.137 mm in thickness.The structure is shown in Fig.3.Detailed information about this structure can be found in Ref.［18］.

Fig.3 A truss bridge model

The FE model of this structure was constructed with the FE program developed in MATLAB environment.Mass of 0.454 kg for each joint block，and 0.1 kg for each collar are considered in FE model， but the momentofinertia ofthese masses are neglected.The modal frequencies and modal shapes，computed by the analytical FE model，are listed in Table 1 and shown in Fig.4 respectively.

Table 1 Analytical and identified modal parameters

Fig.4 Comparison of the modal shapes

5.2 Results of Modal Test

Dynamic test on this structure was performed，and Eigen-system Realization Algorithm(ERA)［19］and Frequency Domain Decomposition (FDD)［20］techniques were applied to identify the modal parameters from the measured accelerations.During the modal test，total 24 accelerometers were used to measure the structuralvibration both in vertical direction and horizontal direction.Six modes of modal parameters were identified，as listed in Table 1.From the results shown in Table 1，greatdiscrepancy between the analytical natural frequencies and those obtained by identification were observed.

5.3 FE Model Updating Based on Meta-Model

5.3.1 Selection of the updating parameters

For the truss bridge structure shown in Fig.3，the structural details except the support condition are believed to be sufficiently modeled in the analytical FE model，therefore，the boundary parameters are to be updated in this study.

As shown in Fig.5，the structure is supported by two pin supports at one end and two rollers supports at the other.At the pinned end，the structure is pinned to a rigid support with bolt，which realizes the constraints of structural movement along the three axes，and the rotations about these three axes are assumed free(see Fig.5(a)).At therollerend，twoconnective components are joggled each other to prevent the structural movement both in the horizontal direction and vertical direction，but allow the free movement in the longitudinal direction of this structure(see Fig.5(b)).

Compared the errors of frequency shown in Table 1，it is easy to found out the big difference lie in mode 1，mode 3，and mode 4.As shown in Fig.4，above three modes of modal shapes vibrate mainly in the horizontal direction，so it is deduced that the main contribution of modeling error goes to the stiffness of the whole structure in the horizontal direction.To verify this deduction，as shown in Fig.5，the complicated configuration of the support connection in horizontal direction is consist with our deduction.Therefore，weselected the connection stiffness of boundary in horizontal direction as the important updating parameters，i.e.，q2，q8，q10as shown in Fig.5.

Fig.5 Detail of support condition

5.3.2 Generation of the meta-model

According to above section， some support constraints are selected as the updating parameters.To realize this in the analytical FE model，a spring element without mass at the DOF of each constraint is applied to model the boundary condition. By comparing different value for couple times，the initial value of each spring stiffness takes 1×1010(N/m)in order to model the rigid connection at each constraint. Finally，we assign a coefficient，whose value locates the interval［0，1］，to the stiffness of each spring element，and these coefficientsare the updating parameters(the value one means the rigid constraint and the free constraint goes to the value zero).

As shown in Table 1，there are no big difference between the measured modal shapes and the analytical modal shapes，so only the first 6 modes of frequencies are deemed as the goal of FE model updating.

Using the method described in Section 3.1，the set of feature samples Γ(80 samples)are generated，and the value of each sample belongs to［0，1］.The corresponding setofresponse function Ζ(Γ)is obtained byusingtheFE modelanalysis.With generated Γand Ζ(Γ)，the meta-model could be built.

Since only the boundary conditions are applied to update the frequency in thisstudy，the second derivative in Eq.(20)is zero.Theoretically，the relationship between the frequency and stiffness of the spring element is lowly nonlinear，so both the RSM and Kriging model are apt to replace this relationship.To compare the performance of Kriging model and RSM，these two meta-model are built by using the same set of feature samples.The RSM takes the second-order polynomial as shown in Eq.(9)，and，for Kriging model，the trend item in Eq.(12)takes the constant and the random item takes the Gauss function.Using the same Γand Ζ(Γ)，all the model coefficients of two meta-modelare obtained. Because the methods described in Section 3.3 cannot be used for RSM，we used 50 extra samples to compare the performance of two meta-model.The comparison of the frequency and modal shapes between Kriging model and RSM are shown in Fig.6 and Fig.7(results of the first three frequencies and modal shapes).Based on these results，it is conducted that the Kriging model is better than RSM for this example.Therefore，the generated Kriging model is applied to update the FE model of this truss bridge model.The RSM and Kriging model were generated by using MATLAB regression package and DACE toolbox(http://www.imm.dtu.dk/～hbni/ dace/)respectively.

5.3.3 Results of FE model updating

In this example，the first 6 measured frequencies are deemed as the goal of FE model updating，and total 3 updating parameters are selected，as discussed in section 5.3.1.Then，the optimization problem shown in Eq.(1)is built，and the procedure described in above section is applied to update the FE model of this truss bridge model.The frequencies of models and their difference before and after updating are listed in Table 2，and it is shown that the updated finite element model and test model match well.The discrepancy of modal frequencies is greatly reduced，and generally a discrepancy after modal updating is less than 3.5%.

Fig.6 Comparison of frequency between Kriging model and RSM

Fig.7 Comparison of MAC value for modal shapes between Kriging model and RSM

The parameters of boundary support before and after FEM updating are listed in Table 3.As shown in this Table，the connection stiffnessin horizontal direction at roller end has the biggest error，which is consistent with the physical condition.The big change of q8represents the horizontal flexibility at roller end (the small gap shown in Fig.7).

Table 2 Comparison of the results before and after FEM updating

Table 3 Comparison of boundaries before and after FEM updating

6 Conclusions

It is an effective way for using the meta-model to improve the efficiency of FE model of large-scale and complicated structures.Two main types of meta-model are introduced in detail，i.e.，Kriging model and RSM.Compared the performance of above two metamodel，some conclusions are drawn as follows.

1)It is point out that there is no random error when the RSM is applied to deterministic FE analysis，so all the methods based on statistical theory cannot be applied to assess the RSM and determine the mathematical form of RSM.This issue does not exist in Kriging model.

2)it is deserve to mention that some tough challenges of FE model updating cannot be improved by using the meta-model except the efficiency issue of large-scale structures.

For the issue of FE model updating，the following advices are summarized in order to select a reasonable meta-model.

1)The determining of a reasonable meta-model depends on the nonlinear degree between the static or dynamic characteristics of structures and the updating parameters，so the selection of meta-model should consider the selected updating parameters and the structural characteristics deemed as the goal of FE model updating.

2)During the assessment of the generated metamodel，the leave-h-out cross-validation is a good choice if it is difficult to build some extra test samples.

3)If the RSM is utilized，all the approached described in Section 3.3 cannot be used at all.The only way to evaluate the accuracy of the generated RSM is to check the error between the predicted response function and the true values with respect to the test samples.

4)To the large-scale and complex structures，e.g.long-span bridges，the relationship between the structuralcharacteristics and updating parameters always is a high nonlinear function，so it is recommend to utilize the Kriging model to the surrogate model of structures.

Appendix A

Assuming the set of feature samples of updating parameters is Γ=(［θ1，θ2，…，θN］T)N×q，and each vector of updating parameters is θv={θv1，θv2，…，θvq}T(v=1，2，…，N)(N is the number of total samples，and q is the number of updating parameters).The corresponding response function obtained by the analytical FE model is Zj(Γ)={Zj(θ1)，Zj(θ2)，…，Zj(θN)}T(j=1，2，…，p，and j represents the number of measured modal parameters or measured displacement etc.).

The procedure of Latin hypercube consists of three parts，i.e.，

(i)Divide the interval of θvinto r non-overlapping intervals having equal probability such as the uniform distribution;

(ii)Sample randomly from a uniform distribution a point in each interval at each dimension;

(iii)Pair randomly(equal likely combinations) the point from each dimension.

Appendix B

According to the content of Section 4，we assume the real Zj(θv)is defined as，

where a (ρj，w，θv)is the random errorterm.Furthermore，the estimation of Ζj(Γ)is supposed as，

where A=［α1，…，αN］Τ，F(xiàn)=［f(θ1)，…，f(θN)］T，B=［β1，β2，…］，Σ=［γ(θ1)…γ(θN)］T.

Using above equation，the^Zj(θv)show in Eq.(23)is rewrote by，

Therefore，the error between^Zj(θv)and Zj(θv)is defined as，

To keep the unbiased estimator，the following equation is obtained，

And then the error defined in Eq.(B4)simplified as，

So the mean square error of the estimation is defined as，

With the definition shown in Eq.(16)，above equation is rewrote as，

Minimizing the Vj(θv)with the constraints of the unbiased estimation，the following optimal objective function is defined as，

where η is the Lagrangian function，and the following results are obtained by

To obtain the coefficients ρjof the correlation function，the following maximum likelihood function is defined as，

where，

With Eq.(B13)，the following results are obtained by solving

Since both β*and var(γ(θv))are the function of ρj，the following optimal objective function is defined as，

Finally，the ρjis obtained by solving above optimization problem，and then the Kriging model is determined.

Appendix C

To assess the accuracy of the generated metamodel，a common way is to compare the error between the true values and estimated response function at extra test samples.The index SST(Total sum of squares)is defined as，

The index SST is composed of two parts，i.e.，SSR(Sum square of regression)and SSE(Sum square of error).

SSR is defined as，

SSE is defined as，

With the SSR and SSE，the following index is defined as，

RS2is applied to evaluate the confidence extent of the generated meta-model，and the closer to 1 the value is，the high confidence the meta-model has.Usually，above procedure needsmany extra test samples，which is difficult to be realized in practice.Therefore，the Leave-h-out［15］cross evaluate strategy is a good choice to reduce the number of test samples.

With the variance analysis，the mathematical form of the meta-model could be determined further.Taken Eq.(11)as an example，if we assume that both SSR and SSE satisfy the Gauss distribution，the following index［16］is defined with the F-test.

where ζ and N-ζ-1 represent the DOF of SSRgand SSEgrespectively.

Since Fs satisfies the F-distribution，if the value of Fsgis biggerthan Fα，ζ，N-ζ-1underthe given confidential level α， this term has significant contribution to the response function，and this term will be kept in the mathematical function or will be deleted if this item has little contribution to the response function.Repeating this procedure，the mathematical form of the meta-model could be determined.The procedure of the generation of meta-model is described as follows.

(i)Select the feature set of samples Γ by using the feature selection method;

(ii)With the FE model analysis，obtain the response variable Ζ(Γ)with respect to selected feature samples;

(iii)Assume the initial mathematical form of meta-model，e.g.，RSM or Kriging model;

(iv)Obtain the coefficient vector β of metamodel;

(v)Determine the test set of samples，and calculate the SSR and SSE by using Eq.(B4)and Eq.(B5);

(vi)Determine the final mathematical form of meta-model by using Fs obtained from Eq.(B6);

(vii)Assessing the accuracy of the generated meta-model by using the method in Section 3.3，if the accuracy is good enough，output the results or go back to step(i).

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