Application of Support Vector Regression (SVR)

Competitive Manufacturing - Proc. of the 2nd Intl. & 23rd AIMTDR Conf. 2008 M.S.Shunmugam and N.Ramesh Babu (Eds) Copyright © 2008 IITMadras, Chennai, India

Application of Support Vector Regression (SVR) in Predicting Spring back in V Bending 1

Apurv Kumar, 1Vishwanath P., 2Amit Kumar Gupta and 3 Swadesh Kumar Singh

(* Corresponding author, [email protected]) Department of Mechanical Engineering, IIT Chennai 2 Department of Mechanical Engineering, BITS Pilani Campus Hyderabad 3 Department of Mechanical Engineering, GRIET, Bachupally, Hyderabad 500072 1

Abstract:-The main application of sheet bending is in automobile industries. Springback depends upon the elastic recovery and extent of deformation that is provided on the material. In bending of thin sheets this causes the angles finally produced to be different from what was intended. Thus exact calculation of Springback has considerable effects on the manufacturing of bent parts with acceptable tolerance limits. In the present research, initially the springback is being calculated in 90 0 V-bended sheets using finite element simulation (FEM). Support vector regression (SVR) is applied on the results of FEM in order to predict the springback in V-bending. Using SVR the die angles are calculated to produce 0 90 bend in the sheet. The results of SVR present a close agreement with the experimental results. Key Words: Bending; Sheet Metal Forming; Springback; Finite Element Method; VBending; Support Vector Regression.

1. INTRODUCTION Metal forming is one of the most important steps in manufacturing to obtain desired shape and size of a material by subjecting it to large plastic deformation [1]. Bending is the plastic deformation of metals about an axis with little or no change in the surface area. In bending process, elasticity limits of materials are exceeded and not the flow stresses. Therefore, the material still keeps a portion of its elasticity. When the load is removed, the material tries to retrieve its original form and the material bent expands backwards with some amount of stretching. This behavior of the material is called Springback. Material parameters such as elasticity, yield stress, hardening property, and process parameters such as the load applied, thickness of sheet metal, die and punch angles, die and punch radii affect Springback in a complex way. With the advent of computation technology, Nilsson et al. [2] and Forcellese et al. [3] performed several research works on the numerical simulation of sheet metal bending process. Prior [4], Finn et al

[5], Huang and Leu [6] reported explicit-implicit solvers to simulate bending processes taking attention to the Springback effect. Zhang et al. [7] and Tan et al. [8] made very interesting contributions to the understanding and simulation of bending processes and their main defects. The Springback increases with increasing punch radius and die opening width and decreasing thickness of sheet metal [8]. It also increases with increase in Spring-back ratio by increasing the punch stroke [3]. Springback value increases when there is an increase in bending angle [7]. In the present research the Springback for extra deep drawn quality (EDD) steel is predicted by using explicit finite element code Dynaform with LSDYNA solver for 900 V-bend and dies and punches were designed to produce 900 bend angle in the sheet after Springback.

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2. FINITE ELEMENT METHOD Extensive knowledge of influence of all the variables on the sheet metal bending process is required to design the tools adequately to manufacture a product with the desired shape and performance. To reduce the time and cost, process modeling by computer simulation is being used to replace the experimental trial and error process by a virtual trial and error process. To analyze the complex problems like that of bending, deep drawing etc , numerical approaches such as the finite element method (FEM) and artificial neural network have been employed rather than analytical methods because of the large number of variables affecting the process. The geometric models of the die, punch and blank are constructed in the pre-processor. Relatively finer meshing is given to the blank to obtain accurate results. The blank used in this case is of size 80x50mm. In the present study, BelytschkoTsay thin shell elements were used for the blank and the tools because of lower computational time 3050% less processing time than the others. The shell thickness was taken as equal to the thickness of the sheet i.e., 1mm. The number and type of elements used in a typical simulation for different objects are shown in Table1.

Fig.1 Springback in Simulation To determine the Springback in V-bending in order to get a blank of 900 bend angle by studying the effect of punch and die angle on the sheet, the punch stroke distance and punch and die corner radii are fixed. A gap equal to the sheet thickness is maintained between the final position of the punch and die to avoid crushing of the sheet. The punch and die corner radii are fixed at ten times the sheet thickness because a smaller radius leads to the formation of cracks and larger radius leads to inaccuracies in the deformed blank.

Table 1: Parameters used in typical FEM simulation TOOL TYPE AND TYPE OF NO. OF BODY ELEMENTS PUNCH 1700 Rigid (Quadrilateral) DIE 2450 Rigid (Quadrilateral) 2640 Deformable BLANK ( Quadrilateral)

3. SUPPORT VECTOR REGRESSION Support Vector Machines (SVM) is a part of Supervised Learning, a branch of statistical learning which learns through a series of examples and gets trained, i.e., it creates a decision-maker system which tries to predict new values. When SVM is applied to regression problems, then it is called as Support Vector Regression (SVR). The prediction of continuous variables is known as regression. Recently, learning and soft computing based approaches such as artificial neural networks (ANNs) and SVRs are widely used in functional regression problems [9]. Although both data modeling methods of ANNs and SVRs show comparable results on the most popular benchmark problems, the theoretical status of SVRs makes them an attractive and promising area of research [10].

The blank material in the present investigation exhibits larger planar anisotropy. So two nodes at every regular position of the blank on either side of the bend are selected and the average of all the included angles between them is taken. The difference of blank angles before and after considering Springback is the Value of Springback. The Springback in the simulation is shown in the Fig. 1.

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Application of Support Vector Regression (SVR)

The SVR was presented as a learning technique that originated from the theoretical foundations of the statistical learning theory and structural risk minimization. The SVR first nonlinearly transforms the original input space x into a higher dimensional feature space. That is, in order to learn nonlinear relations with a linear machine, it is required to select a set of nonlinear feature and to express the data in the new representation. This is equivalent to applying a fixed nonlinear mapping of the data to a feature space which the linear machine can be used in. This transformation can be achieved by using various nonlinear mapping. Nonlinear regression problems in input space can become linear regression problems in feature space.

weight vector norm. An increase of the regularization parameter penalizes larger errors, which leads to a decrease of approximation error. This can also be achieved easily by increasing the weight vector norm. However, an increase in the weight vector norm does not make sure of the good generalization of the SVR model. The constants and are user-specified parameters and |yi f(x)| is called the -insensitive loss function [11]. The loss equals zero if the predicted value f(x) is within an error level , and for all other predicted points outside the error level , the loss is equal to the magnitude of the difference between the predicted value and the error level (refer to Fig. 2 and 3). In SVR modeling, the data points which lie on the margin lines ( y f (x ) ) are the support samples or support vectors or support set, whereas the data points which lie inside the margin lines are called as remaining set and the data points which lie outside the margin lines are called error set. Increasing the insensitivity zone means a reduction in requirements for the accuracy of approximation and it also decreases the number of support vectors, leading to data compression. In addition, increasing the insensitivity zone has smoothing effects on modeling highly noisy polluted data.

The SVR model is given N training data N xi , y i i 1 R m R , where xi is the input vector to the SVR model and yi is the actual output value, from which it learns the inputoutput relationship. The SVR model can be expressed as follows [7]: N

y

f ( x)

wi i ( x) b

wT ( x) b (1)

i 1

where the function

i

(x) is called the feature that is

nonlinearly mapped from the input space x,

w [ w1 w2

w N ]T , and

[

1

2

N

]T .

Eq. (1) is a nonlinear regression model because the resulting hyper-surface is a nonlinear surface hanging over the n-dimensional input space. However, after the input vectors x are mapped into vectors (x ) of a high dimensional kernel-induced feature space, the nonlinear regression model is turned into a linear regression model in this feature space. The nonlinear function is learned by a linear learning machine where the learning algorithm minimizes a convex functional. The convex functional is expressed as the following regularized risk function, and the parameters w and b are a support vector weight and a bias that are calculated by minimizing the risk function: N 1 T R ( w) w w y i f ( x) e (2) 2 i 1 where 0 if y i f ( x ) y i f ( x) e y i f ( x) otherwise

Fig. 2: Linear -insensitive loss function

The constant is called a regularization parameter. The regularization parameter determines the trade-off between the approximation error and the

Fig. 3: Non-linear regression with -insensitive band in the SVR model

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Parrella [12] developed an Online SVR toolbox for SVR modeling in MATLAB application. This toolbox is used here for the application of SVR in predicting the spring back in bending operation. The input parameters have been normalized between 0 and 1. The training set X is the combined vector of the four input parameters (Punch and die angle and punch and die corner radius). The range of input parameters is shown in Table 2. The training set Y is the response parameter, i.e., spring back.

parameters as discussed before blank angle after spring back is being calculated using simulation. Using SVR punch and die angles were calculated to produce bend of 900. It was found that for die angle of 880 and punch angle of 900, the bend angle after spring was 90.240. With the same design simulation was performed and it was found that the blank angle after spring back is 90.0440. This shows that SVR is a very effective tool in decreasing the number of iteration in the sheet metal forming operation.

Table 2: Input parameters and their range for SVR modeling Input parameters Punch angle Die angle Punch corner radius Die corner radius

Range 85-950 85-950 3-15 mm 3-15 mm

SVR model is trained using 24 sets of inputoutput pairs calculated by simulation. Table 3 represents the training set data found out by simulation in the present research. Training parameters are initialized as: cost function (C) = 10; -insensitive loss function ( ) = 0.01; kernel type = Radial Basis Function (RBF); kernel parameter = 30. SVR trains the data one by one by adding each sample to the function simultaneously checking if the Karush-Kuhn-Tucker (KKT) conditions are verified. If the KKT conditions are verified the sample is added, or else the sample is stabilized using the stabilization technique. The stabilization technique dynamically changes the SVR parameters, cost function and epsilon insensitive loss function to optimize the values.

Fig. 4: Blank angle after Springback with 87.8550 Die and 900 Punch The optimum dimensions of the punch and die were selected based on the 900 bend and minimum variation in thickness and VonMises stresses. So the punch angle of 900 and die angle of 87.8550 respectively were selected.

4. RESULTS AND DISCUSSIONS

Die and Punch are manufactured to verify the spring back predicted by SVR. Tooling was arranged on a 40 ton UTM as shown in fig 6. Five samples of the sheet were deformed to have consistency in the results. The deformed blank is kept on the surface plate to measure the angle using the bevel protractor of 5 minute accuracy. The comparative results of the study of Springback are given in Table 4.

Initially the punch and die angles were fixed at 900 and the analysis was run using the FE code. The deformed blank from the post-processor is analyzed and the bend angle is calculated as shown in Fig 4 and found that the blank was bent to an angle of 92.1450 and that there was a 2.1450 Springback in the sheet. The die angle was then varied by keeping the punch angle to be at 900. In the first iteration the die angle was taken to be 87.8550 by deducting the Springback value which was calculated from the original die angle. By varying the different suddenly increases when the deformation of the material is about to complete. The force rapidly

The load curve found in the experiment is shown in Fig. 5. It can be seen in that load comes on the sample during initial bending operation and it increases linearly from zero. This is mainly due to the elastic behavior of the material. The rise in the load

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requirement is due to the developing and spreading of the plastic zone whereas the decrease is due to the large rigid body displacement and lower deformation resistance from the specimen. The load suddenly

increases when the bending process is about to complete. The increase in load is mainly due to close proximity of punch with die

Table 3: Training set for SVR

Sno.

PUNCH ANGLE

DIE ANGLE

PUNCH RADIUS

DIE RADIUS

BLANK AFTER SPRINGBACK

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

90 90 90 90 90 90 90 90 90 90 90 90 90 89.188 89.757 90 90 90 90 90 90 90 90 90

87.044 87.71 87.855 87.87 88.84 88.96 89.19 89.89 90 91.13 91.16 91.39 92.92 87.855 87.855 90 90 90 90 90 90 90 90 90

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 9 10 11 12 13 14 8

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 9 8 8 5 5 5 5 5 10

89.058 91.571 90.044 88.35 89.927 90.071 90.447 90.447 92.145 94.132 94.07 95.423 94.503 89.116 89.079 90.178 94.322 92.197 92.092 93.118 94.632 94.396 93.308 92.736

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thickness the %error varies from 5-25%. The reason why it results in reduced effort is because it predicts the springback in quite proximity to experiment. Also significant reduction in the springback prediction can be done by increasing the input data values for which the SVM gets trained.

Table 4 Comparison between Simulated Results and Experimental Results. Results Die angle from simulation Angle of die manufactured Punch angle from simulation Angle of punch manufactured Blank angle from simulation Blank angle from SVR Blank angle from experiment Springback from simulation Springback from experiment

Value 87.8550 880

.

900 900 90.0440 90.240 91.40 2.1890

. Fig.5: Punch load Vs displacement curve during the bending of the sheet

3.40

5. CONCLUSIONS The die and the punch in the present investigation were designed using FEM and SVR techniques to get a bend angle of 90o in the sheet and also to estimate the amount of Springback that a material undergoes during deformation. It was found from the simulation that, the Springback for this particular material is 2.189o with the optimum punch and die. The Springback of 3.40 was observed in the experimental investigations. It was found that the pattern of Springback was well predicted using FEM code and SVR technique and exact bends can be provided in the materials by designing the dies using FEM. This approach can be scaled up to other geometries namely U bending or flanging where springback prediction is significant in order to achieve accurate products. As the FE model was used ,other geometries can be similarly modeled and SVR prediction method can be used to predict springback for different materials and geometries. However the same model cannot be used as the data values correspond to a particular geometry which was optimized to reduce springback in V-Bending such as the punch radius , die radius etc were optimized using FEM.SVR is a useful tool in obtaining prediction when the data are not regularly distributed or have unknown distribution. In case of bending as Nilson.et al[2] has shown in predicting the springback using FEM for different material and varying sheet

Fig. 6 Showing the assembly of tools on 40 ton UTM machine

6. REFERENCES

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1.

Dieter, G. E., Mechanical Metallurgy, 1988, McGraw Hill, Singapore.

2.

Annika Nilsson, Lars Melin, Claes Magnusson, Finite-element simulation of V-die bending: a comparison with experimental results, Journal of Materials Processing Technology, 1997, 65(1-4), 5258.

3.

Forcellese, A., Fratini, L., Gabrielli, F. and Micari, F., Computer aided engineering of the sheet bending process, Journal of Materials Processing Technology, 1996, 60(15), 225232.


4.

Prior, A. M., Applications of implicit and explicit finite element techniques to metal forming, Journal of Materials Processing Technology, 1994, 45, 649656.

5.

Finn, M. J., Galbraith, P. C. and Wu, L., Use of a coupled explicit-implicit solver for calculating spring-back in automotive body panels, Journal of Materials Processing Technology, 1995, 50, 395409.

6.

Huang, Y. M. and Leu, D. K., An elastoplastic finite element analysis of sheet metal U-bending process, Journal of Materials Processing Technology, 1995, 48, 151157.

7.

Zhang, L. C., Lu, G. and Leong, S. C., Vshaped sheet forming by deformable punches, Journal of Materials Processing Technology, 1997, 63(1-4), 134-139.

8.

Tan, Z., Persson, B. and Magnusson, C., An empirical model for controlling Springback in V-die bending of sheet metals, Journal of Materials Processing Technology, 1992, 34(14), 449455.

9.

Kulkarni, A., Jayaraman, V. K. and Kulkarni, B. D., Control of chaotic dynamical systems using support vector machines, Physical Letters A, 2003, 317, 429435. 10. Kecman, V., Learning and Soft Computing, 2001, MIT Press, Cambridge, MA. 11. Vapnik, V., The Nature of Statistical Learning Theory, 1995, Springer, New York. 12. Parrella, F., Online Support Vector Regression, A thesis presented for the degree of Information Science, 2007, University of Genoa, Italy.

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