Prediction of Springback in the Air Bending Process Using a Kriging Metamodel

This paper addresses the use of the kriging approach to predict the springback in the air bending process. The materials and the geometrical parameters, which significantly affect the springback, were considered as inputs, and the springback angle was considered as the response. A verified nonlinear finite element model was used to generate the training data required to create the kriging metamodel. The training examples were selected based on computer-generated D-optimal designs. A comparison between the kriging approaches and the response surface methodology is conducted and discussed. The results showed that kriging accurately predicts the finite element springback results. Comparing the accuracy of kriging with a response surface methodology shows that kriging with a 2nd degree polynomial and exponential correlation function predicts the springback more accurately than the response surface methodology. Keywords-Metamodels; Springback; Kriging; Response Surface Methodology; D-Optimal Designs

A number of analytical models based on the geometry and the material characteristics have been conducted using the analytical methods for springback predictions.The recent approaches to analytical solutions include the prediction of springback considering Young's modulus variation with a piecewise hardening function [23], friction modeling of high strength steels [24][25], semi-analytical modeling of springback [26], U-bending as a function of stress distribution in the thickness of sheet metal [27], springback prediction of an asymmetric thin-walled tube [28], and an analytical springback model for lightweight hexagonal close-packed sheet [29].In general, most of the analytical models assume a simplified process and material properties due to the complexity of the problem.
The finite element method is a powerful tool for analyzing sheet bending processes with various materials and test conditions [1,11,[30][31][32].Because a large number of FEMs is required to develop an accurate approximation for springback prediction, it is not practical, relatively time-consuming, and computationally expensive to apply it for a complex process, such as metal forming.In addition, a direct link between finite element code and optimization routines can lead to inefficient results because a large number of iterative calls are required during the optimization.Recently, there has been a great increase in the application of metamodels instead of the complex analytics models that are limited by assumptions [33][34].Several metamodeling techniques with various degrees of complexity have been extensively applied, such as the response surface methodology [35][36][37][38], artificial neural network [39][40][41][42], radial basis function [43][44][45], and kriging [46][47][48][49].Some of these techniques are suitable for global approximations, i.e., can be used for representing the complete design space, while others are more suitable for local approximations of a part of the design space.The existing studies to predict springback in the air bending process mostly use training data from experiments with real-life systems.Consequently, these studies considered only inadequate materials and tool geometry [37,[39][40][41]50].
In this study, the kriging approach is developed to predict springback in the air bending process.The finite element simulation is employed to generate the training data required to train the kriging metamodel.The use of a finite element analysis permits the mapping of a large range of materials and geometrical tool dimensions.The kriging metamodel is compared with the traditional response surface methodology (RSM) to show the merits of the kriging metamodel to predict springback.

II. METAMODEL GENERATION
A metamodel [46] is a mathematical approximation of a computationally expensive numerical model.Using a metamodel, one can study the behavior of the response function more efficiently and can perform an optimization very rapidly.The mathematical approximation can be written as:

A. Response surface methodology
RSM [35] is a set of mathematical methods for enhancing processes and products.The relationship between the response (y) and the vector independent factors (x) can be expressed by: where ε is the random error, which is expected to be normally and independently distributed with a mean of zero and a standard deviation σ.The broadly used RSM functions are loworder polynomials and can be expressed as: For considerable curvature, a second-order polynomial can be expressed as: The parameters in (4) and ( 5) are typically determined by a least squares regression analysis.The least squares estimators of the regression coefficients b are denoted by b  . The regression model may be written as:  Thus, Y is a vector of n responses, X is an n × p design matrix consisting of variable settings, β is a vector of p regression variables and ε is a vector of the n errors.In the design matrix, each row corresponds to one design point, and each column corresponds to one regression coefficient.The fitted regression model becomes: Assuming that β has a zero mean and a variance-covariance matrix given by n I 2  , the so-called ordinary least-squares estimator of β is [38]:

B. Kriging metamodel
The kriging interpolation method [47], developed in the field of spatial statistics and geo-statistics, has been widely used in recent years to model deterministic simulations.Kriging predicts the value of a new point using stochastic processes.Kriging consists of a combination of a known polynomial plus fitting errors, as shown in 10: where Y(x) is the unknown response of interest, is the (P×1) vector of regression functions, and is the (P×1) vector of unknown coefficients.
T f (x)β globally approximates the design space.In many cases, T f (x)β is assumed to be a constant (ordinary kriging), or it may be a first-order or second-order polynomial."Z(x)" is a stochastic process, which has a mean of zero, variance of σ 2 , and the following covariance matrix:  Z(X) performs localized deviations, so that the kriging model interpolates n s sampled data.The correlation matrix R is n s × n s is symmetric with ones along the diagonal.[R(S i , S j )] is a user-defined correlation function.The selection of the correlation function determines on how to fit the data.In this study, the most popular correlation function is Gaussian, in which the exponent 2 is used in (12).

( , ) exp
The quantities are the k th components of the sampled points S i and S j , respectively.As part of the kriging approximation, the unknown θ k terms are determined.The maximum likelihood estimation of θ k can be calculated by maximizing the following expression: s  is the estimated process variance and can be calculated by: The kriging estimate at any untried point x is given by: www.etasr.comKhadra and El-Morsy: Prediction of Springback in the Air Bending Process Using a Kriging Metamodel ( ) F is a column vector with n s components, which is filled with ones when p(x) is a constant.The procedure to predict springback in the air bending process consists of the following phases.A schematic representation of the methodology for springback angle prediction using a kriging metamodel is shown in Figure 1.Flowchart of the methodology for springback prediction using kriging metamodel.

A. Identify the design space and select the design variables
The first step is to select the design variables (E, σ y , K, n, t, R p , R d , W d , θ 1 ), which is a very important step for generating accurate models (Table I).The method for selecting the sample points, which is dependent on the design variables, is called the design of experiments (DOE).These sample points are used as the input variables for the next step.In this study, the Doptimality experimental design criterion [51] was used to sample the design space.D-optimality is the most common criterion, which seeks to maximize |X T X|, the determinant of the information matrix (X T X) of the design.Equation 18shows the selection of X T out of all possible design matrices chosen from ξ n .
Maximizing the determinant of the information matrix (X T X) is equivalent to minimizing the determinant of the dispersion matrix (X T X) −1 .

B. Generate the training points using FE simulation
The first step in generating the training points is to build an accurate FE model of the air bending process using the design space obtained from the previous stage.Figure 2 shows a schematic of FE model.The material is assumed to be isotropic and homogeneous.The FE analysis is simplified to a 2D plane strain.A 4-node, iso-parametric, arbitrary quadrilateral plane-strain element is applied.This element uses bilinear interpolation functions by activation of the assumed strain formulation.The strain hardening is defined by Swift's Eq. which is given by: The equivalent plastic stress s is described by Swift's exponential hardening law, based on the material's consistency K, the equivalent plastic strain e , and the hardening exponent n.The input parameters influencing the amount of springback are the modulus of elasticity, the strain-hardening coefficient n, the yield strength σ y , and the geometrical parameters.The geometrical parameters include the thickness of sheet t, radius of punch R p , radius of die R d , width of die W d , and bending angle θ1.The developed metamodel can be represented by: 1 ( , , , , , , , , )

C. Metamodel building
In this study, two metamodel techniques were used, KMM and RSM.Two training datasets were generated using the FE simulation stage.The first dataset is based on a linearinteractions (LI) model.A model for two variables can be given by: The second dataset is based on the full quadratic (FQ) model.For example, a quadratic model for two variables is For validation, the metamodels use a dataset, which is different from the above selected datasets to create the metamodels.The study was performed using a combination of selected datasets and different regression functions for kriging to find the best metamodel for the prediction of the springback angle in the air bending process.The results are presented for the following kriging metamodels: • Kriging metamodel with a constant and an exponential correlation function (P0 EXP ) • Kriging metamodel with a 1 st -degree polynomial and an exponential correlation function (P1 EXP ) • Kriging metamodel with a 2 nd -degree polynomial and an exponential correlation function (P2 EXP )

D. Metamodel validation
The accuracy of the metamodel is influenced by the metamodel type as well as the quality and quantity of the training dataset.The metamodels should be validated before being used as substitute models by comparing the metamodel results.In this study, useful means to determine the accuracy and the predictive capability of the metamodels are the normalized root-mean-squared error (NRMSE), the normalized maximum absolute error (NMAX), and the statistical correlation (R 2 ).
( ) where Y and Y  are measured and predicted values.
Additionally, Y  is the mean of the actual function values at m test points.The larger the R 2 , the more accurate the metamodel.The lower NRMSE and NMAX, the more accurate the metamodel.The R 2 and the NRMSE provide a measure of the overall accuracy, while the NMAX is a measure of the local accuracy of the model.To provide a reliable accuracy, the validation set must be large enough and spread out over the design space.

IV. DISCUSSION OF THE RESULTS
Figure 3 shows the training points that are space-filling for two factors, the yield strength and the bending angle before springback θ1.From the histogram figures, the improvement in the space filling capabilities can be observed when the FQ model is used.The selected design points are closely and uniformly distributed through the design space.Table II summarizes the coefficient of determination (R 2 ) for the full quadratic and linear interaction models From the results in Table II, it can be observed that the model in which examples were selected based on the FQ model shows a higher correlation coefficient when compared with the model in which examples were selected based on the LI model.
Finally, the model with examples selected based on the FQ model reached a significant level (0.0012 in the case of P2 EXP ), which is more reliable than the model based on the LI model (0.026 in the case of P2 EXP ).Therefore, the FQ model shows a better fit when compared to the LI model, and the R 2 values for the full quadratic models are satisfactory.Figure 4 shows the relationship between the normalized error and the metamodel type for both the fitted LI (Fig. 4a) and FQ (Fig. 4b) models.The relationship between normalized error and metamodel type for both trained models.
Figure 5 shows the result of the metamodel-predicted springback angle versus the actual FEM.Fig. 5a, 5b and 5c refer to the KMM of P0 EXP , P1 EXP and P2 EXP , respectively.Fig. 5d refers to the RSM model.For accurate prediction, all of the points should lie on the 45° line.In the case of P0 EXP and P1 EXP , the prediction of the kriging metamodel is not accurate compared with P2 EXP and RSM, as shown in Figs.5c and 5d.
In the case of RSM and P 2EXP , all of the points are close to the 45° line.The lines representing ± 5% accuracy are drawn to show the percentage error in the prediction.From Fig. 5c, it can be observed that all of the predictions are within ±5% accuracy.The maximum absolute error in the prediction of springback is 2.8%.Fig. 5d shows the best RSM predicted springback versus the simulated FEM prediction.It can be observed that out of 50 predictions, most of the predictions are within ± 5% accuracy; seven predictions are within ± 10% and one prediction is outside the ±10% accuracy.

V. CONCLUSIONS
In this study, the kriging metamodeling approach was implemented to predict the springback angle for the sheet metal air bending process.A comparative analysis of different metamodel schemes based on the kriging predictions has been conducted.On the basis of the computational results, the metamodels proposed avoid a great number of time-consuming evaluation processes and have significant merit in the aspect of accuracy.The kriging metamodels selected using a D-optimal design based on the full quadratic model are able to provide more accurate prediction of the springback than the metamodels based on the linear-interactions model.The kriging metamodel with a 2nd-order polynomial trend function and an exponential correlation function provides a minimal normalized error compared with alternative metamodels investigated in this paper; this error compares the prediction of the springback angle by the metamodels and the finite element simulation outputs.The springback angle of sheet-metal of length n s , which contains the response values at each sampled point.III.METHODOLOGY OF SPRINGBACK PREDICTION USING A KRIGING METAMODEL
Figure4shows the relationship between the normalized error and the metamodel type for both the fitted LI (Fig.4a) and FQ (Fig.4b) models.Fig. 4a refers to the NMAX and Fig. 4b refers to the NRMSE.The accuracy of all of the kriging metamodels and the RSM model is greater in the case of selecting examples based on the FQ model than on the LI model.There are large differences in the normalized error between the FQ model and the LI model.From Fig. 4a, it can be observed that when selecting examples based on the FQ model the best model with the minimum NMAX is the P 2EXP followed by the RSM.

TABLE I .
THE DESIGN SPACE CONSIDERED IN THIS STUDY