Sheet forming optimization based on least square ...

Int J Mater Form (2010) Vol. 3 Suppl 1:9 –12 DOI 10.1007/s12289-010-0694-3 © Springer-Verlag France 2010

Sheet forming optimization based on least square support vector regression and intelligent sampling approach Hu Wang 1*, Guangyao Li2 1

The State Key Laboratory of Advanced technology for vehicle design and manufacture, College of Mechanical and Automotive Engineering, Hunan University, Changsha, 410082, China, [email protected] 2 The State Key Laboratory of Advanced technology for vehicle design and manufacture, College of Mechanical and Automotive Engineering, Hunan University, Changsha, 410082, China, [email protected] ABSTRACT: In this paper, a metamodel-based optimization method by integration of support vector regression (SVR) and intelligent sampling strategy is applied to optimize sheet forming design. Compared with other popular metamodeling techniques, the SVR is based on the principle of structure risk minimization (SRM) as opposed to the principle of the empirical risk minimization in conventional regression techniques. Thus, the accuracy and robust metamodel can be obtained. The intelligent sampling strategy is a kind of design of experiment (DOE) essentially. The characteristic of this method is to generate new sample automatically by responses of objective functions. Compared with traditional DOE methods, the number of samples isn’t constant according to different cases. Furthermore, the number of samples and size of design space can be well controlled according to the intelligent strategy. To minimize both objective functions of wrinkling, crack and thickness deformation efficiently, the proposed method is employed as a fast analysis tool to surrogate the time-consuming finite-element (FE) procedure in the iterations of optimization algorithm. An example is studied to illustrate the application of the approach proposed, and it is concluded that the proposed method is feasible for sheet forming optimization. KEYWORDS: Optimization, Least square support vector regression(LS-SVR), Intelligent sampling, Sheet forming

1 INTRODUCTION In order to improve the efficiency of sheet forming optimization, metamodeling technique is widely employed in this field recently. The metamodeling technique is developed as “surrogates” of the expensive simulation processes in order to improve overall computation efficiency. The most widely used metamodeling techniques, such as polynomial regression (PR), Kriging (KG), radial basis function (RBF), multivariate adaptive regression splines (MARS) are widely used for engineering problems are based on empirical risk minimization (ERM) without considering structure risk minimization(SRM). Thus, the generalization ability of the ERM-based metamodels can’t be guaranteed. It is thereby difficult to obtain robust solutions for nonlinear problems. Recently, a new model called support vector regression (SVR) based on SRM and ERM was proposed and tested [1, 2]. Least squares support vector regression (LS-SVR) is one of useful SVR version which involves equality instead of inequality constraints and works with a least squares cost function. In this way, the solution follows from a linear Karush–Kuhn–Tucker (KKT) system

instead of a quadratic programming problem. This reformulation greatly simplifies the problem in such a way that the solution is characterized by a linear system. In this paper, a LS-SVR based metamodeling technique is suggested and applied for sheet forming optimization. To improve the efficiency and accuracy of optimization, an intelligent DOE strategy is adopted at the stage of DOE. The LS-SVR is implemented to construct robust metamodel. Finally, particle swarm optimization (PSO) is used to obtain solutions based on the constructed metamodel. The rest of this paper is organized as follows. The basic theories of LS-SVR and intelligent sampling strategy are introduced in Section 2 and 3 respectively The application of sheet forming optimization is described in Section 4. Section 5 gives the final conclusions.

2 LS-SVR THEORIES Consider a problem of approximating a set of N samples xl , y l l 1 with input data xl and output data N

with yl and corresponding response as presented in Eq.(1)

____________________ * Hu Wang: 410082, 86-731-88821445, 86-731-8822051, [email protected]

10

D   x1 , y1 ,  x2 , y2   xl , yl   xN , y N ,

(1)

xl  R , yl  R The optimization problem in primal weight space can be expressed as 1 1 N (2) Min J ( w,  )  wT w     l2 w, 2 2 l 1 subjected to (3) y l  wT  ( xl )  b   l , l  1,2, N n

with  ()  R  R a kernel that maps the input space into a so-called higher dimensional (possibly infinite n

nh

dimensional) feature space. Weight vector w  R in primal space, error variable  l  R and bias term is b. nh

The cost function J consists of a sum square errors (SSE) fitting error and regulation term. The relative importance of the ERM and SRM terms is determined by the positive constant γ. The model of primal space can be presented as follows (4) y ( x)  wT  ( x)  b The weight vector w can be of infinite dimension, which makes a calculation of w from Eq.(2) impossible in general. Therefore, one computes the model in the dual space instead of the primal space. Then, the Lagrangian multiplier expression applied for Eqs.(2-3) is obtained as N



( w, b,  , a)  J ( w,  )   al wT ( xl )  b   l  yl

 (5)

l 1

with the Lagrangian multipliers al, the criteria satisfied by the optimal solution can be written as N   ( w, b,  , a)   0  w   a l  ( xl ) 

w l 1  N   ( w, b,  , a)  (6)  0   al  0  b l 1    ( w, b,  , a)   0  a   l l      ( w, b,  , a)   0  wT  ( x )  b    y  0 l l l  a

These conditions are similar to standard SVM optimality conditions, except for the condition al=γεl. After elimination of w and ε, the solution is obtained as 0 I T  b   0  (7)  1    I    I  a   y    where (8.1) y  [ y1 , y 2 ,  y N ]

I  [1,1,1] a  [a1 , a 2 , a N ]

(8.2) (8.3)

i , j   ( xi ) T  ( x j ) for i, j  1,2,  N

(8.4)

Based on the Mercer’s condition, there exists a mapping φ(·) and an expression can be written as (9) K ( x, y )    i ( x ) T  i ( y ) , i



2

If and only if, for any g (x ) such that g ( x) dx is finite, one has

 K ( x, y) g ( x) g ( y)dxdy  0 As a result, the kernel K(·,·) such that K ( xi , x j )   ( xi ) T  ( x j ), for i, j  1,2, N

(10) (11)

The final LS-SVR model for function estimation is obtained as N

y ( x )   a l K ( x, x l )  b

(12)

l 1

where al and b are the solutions of Eq.(7). In this study, we focus on the choice of RBF kernel K ( xi , x j )  exp( xi  x j

2

 2 ) for the sequel.

3 INTELLIGENT SAMPLING METHOD The training sample set for the LS-SVR is commonly generated by neural network (NN). If the NN is used to generate samples, the size of training samples is difficult to control, especially for large scale nonlinear problems. Thus, the efficiency of the LS-SVR is no longer advantages compared with other metamodeling techniques. Hence, an efficient boundary and best neighbour sampling (BBNS) [3] strategy is used to generate samples. For practical engineering problems, intervals of design variables can be given according to engineering experiences. And then, the initial samples should be automatically generated by popular DOEs, such as full factorial (FF), D-optimum (D-opt), central composite design (CCD), Latin hypercube sampling (LHS), et al. To save computational cost of the sampling procedure, the initial samples should be sparsely distributed in the design space. When the evaluation procedure (such as FE simulation) is completed, several better samples addressed as “better sample set” are collected to generate new samples by the BBNS principle. The details of the BBNS are described as follows: Step 1. The LHS is implemented to generate the initial sparsely distributed samples in given design space; Step 2. The preliminary design evaluations are performed with the initial samples; Step 3. The several better samples are collected and used to generate new samples; Step 4. The new samples are generated by Eqs.(13) Step 4.1 The value of the new sample is given by

( Nearest ) ( Nearest )  v Current  v1Boundary   v Current  v1Best  , 2 ,n , 2 ,n (13) c1   1, 2,n c 2 v1, 2,n   1, 2,n     m m     where v is the vector of samples, n denotes the size of as default value; for 2 dimensional problems, the default design variables. The distance between the boundary and value is 2), m is inversely proportional to the distance best sample closest to assigned samples can be divided from the current assigned sample, c1, c2 denote the acceleration weight coefficients. They can be determined into m segments specified by the user (m is set equal to n

11

by Eq.(14) according to the response value derived from



the evaluations.

    



Boundary ( Nearest )  R v1Current , 2 ,n  R v1, 2 ,n  c 1 Boundary ( Nearest ) ( Nearest )  R v1Current  R v1Best R v1Current  , 2 ,n  R v1, 2 ,n , 2 ,n , 2 ,n  c2  1  c1



 

where R(·) denotes the response value obtained from the forward simulation with the corresponding sample, such Best

Boundary ( Nearest )

as v1, 2,n , v1, 2,n

Best ( Nearest )

and v1, 2,n

.

The meaning of superscripts is as follows: Current :the current sample in the better sample set; Nearest : the nearest sample from the current sample; Boundary: boundary of the intervals (constraints); Best: the sample which has the best response values of evaluations; Boundary (nearest): the nearest boundary sample from the current sample; Best (nearest): the nearest sample of better sample set. Compared with other popular intelligent sampling methods, the distinctive characteristic of the BBNS is to use the boundary and neighbour information of the design space to generate the new sample dynamically. Step 4.2 If the position of the new samples have been generated before or locate outside of the given intervals, the previous best sample should be substituted by the current sample, and the procedure goes back to 4.1; Step 4.3 The evaluations are performed with the new generated samples; Step 4.4 The better sample sets are updated. Step.5 If Best Best R v new  R vold (15)   ,  (0,1) Best R v new



   



then procedure ends, else it goes to step 3, where η is the threshold which can be set by the users, with the default value in this study given as 10%. The major characteristics of the BBNS strategy are summarized as: 1. The new samples are determined by the responses of the evaluations with previous generated samples; 2. The local convergence can be avoided due to given constraint conditions.

4 APPLICATION In order to predict value of wrinkling and crack, the objective functions based on thickness and forming limit diagram (FLD) are calculated by the numerical simulation results. The first objective function of sheet forming problem is based on thickness variation between the initial and final state and is given by Barlet et al.[4]. 1

k

 hi  hi  k  n (16) f h    f hi  with f hi   e i 0   i 1   h0  where h0i , hei are the initial and final thicknesses of the ith element, n denotes the number of elements, the coefficient k=2, 4, 6 . . . is introduced to emphasize the extremes of the objective function.

 



(14)

In order to evaluate possibility of the wrinkling, we define two FLCs in the principal plan of logarithmic strains proposed by Hillman and Kubli [5]. (17)  1   s ( 2 ),  1   w ( 2 ) where φs is to use control the crack phenomenon, φw is used to control the wrinkling. Both of them depend on the material; they are generally given as knots data in tables. Then, they are defined by Eq.(18)  s ( 2 )   s ( 2 )  s1 (18)   w ( 2 )   w ( 2 )  s 2 where s is called the safety tolerance, take from the true FLC, θs(ε2) and θw(ε2) are the bounds of safety domain. This tolerance is constant during the optimization process and defined by the engineers. Therefore, the second objective function fε is defined for each element (ε1, ε2) by the tolerance between the actual strainε1 obtained by computation and the safety FLC for a given strain ε2. 1

 n k f     f i  with  e 1  (19)  f i   e   ( e ) k for  e   ( e ) s s 1 2 1 2   k  i e e e e f ( ) for (            w w 1 2 1 2) f i 0 otherwise   where the coefficient k=2, 4, 6 . . . is introduced here to emphasize the extremes of the objective function. The multi-objective function given in Eq.(20) is used in this work. (20) f  w1 f h  w2 f

 

   

 

in which,w1 and w2 are weighting factors and w1+w2=1, fh and fε are the square mean values in each design variables’ condition. In this work, w1 and w2are set to 0.5, k is set to 4. The FE mesh consists of 4872 nodes and 4673 shell elements. The material used to produce this part is a high carbon stainless steel, modelled by an isotropic hardening of Krupkowski-Swift as   567.29( p  0.007127) 0.2637 MPa . The transversal anisotropy is taken into account for this material with an averaged Lankford coefficient r  1.77 .The principal geometrical and material characteristics of the blank are given in follows: Young’s modulus, E=206GPa, Poisson coefficient, v=0.3, Material density, ρ=7800 kg/m3, Initial thickness h0=1.00mm, Friction coefficient, μ= 0.144. In this work, the drawbead location lines are shown in Figure 1. The restraining force per millimetre of each drawbead is defined as design variable named as dbf1, dbf2, dbf3. The corresponding constraints are listed as:

12 dbfi  [50,200]( N / mm), i  1,2,3 , dbf j  dbf 2 j  1,3



The constraint of another design variable blankholder force is given as bhf  [200,500] KN.

a: Die

b: Punch

Figure 1: Location of drawbeads

After 8 iterations, the finial optimum is obtained based on 32 expensive and 26 cheap samples. The optimum drawbead restraining forces are (92, 92,183) N/mm and blank holder force is 432KN.To validate the optimized results, the optimum design variables are calculated by the FE method. Figure 2 presents the FLD of finial formed blank with the initial and optimum design variables respectively.

a: The FLD with the initial variables

c: Formed product Figure 3: The real mould and product

5 CONCLUSIONS The advantages of the proposed method are summarised as follows. 1. Due to consideration of the SRM, the LS-SVR can obtain more robust optimum in practical engineering problems; 2. The intelligent strategy BBNS is integrated and make the optimization procedure easy to converge; 3. The proposed optimization method is proved to be feasible for sheet forming optimization.

ACKNOWLEDGEMENT This work is supported by the National Natural Science Foundation of China (NSFC) under grant number 10902037. b: The FLD with the optimum design variables Figure 2: The FLD comparison between before and after optimization

The FLD obtained at the end of the optimization process (with the initial drawbeads restraining forces) is shown in Figure.2 (a). Clearly, several points are found above the “safety FLC”. It means that a crack should occur. The FLD with optimum design variables is illustrated in Figure 2(b). There is no crack risk point. Although there are still several wrinkle points out of “safety FLC”, all these points are out of product segment and will not influence the quality of the product. Thus, The optimum solutions are applied for real engineering application and corresponding mould and product are illustrated in Figure 3(a,b) and Figure.3(c)respectively. Therefore, it is easy to prove that the proposed method is feasible for sheet forming optimization problems.



According to engineering experience, the restraint force on inner drawbead should be larger than outer drawbead.

REFERENCES [1] Clarke, S. M., Griebsch, J. H. and Simpson, T. W., Analysis of support vector regression for approximation of complex engineering analyses, Transactions of ASME, Journal of Mechanical Design, 2005;127 (6),1077-1087. [2] Hu Wang, Enying Li and Guang Yao Li, The least square support vector regression coupled with parallel sampling scheme metamodelling technique and application in sheet forming optimization, Materials and Design, 2009;30(5):1468-1479. [3] Hu Wang, G. Y Li., Enying Li, Z. H. Zhong, Development of metamodeling based optimization system for high nonlinear engineering problems, Advances in Engineering Software, 2008, 39(8):629-645. [4] Barlet O., Batoz J.L., Guo Y.Q., Mercier F., Naceur H., Knopf-Lenoir C., The inverse approach and mathematical programming techniques for optimum design of sheet forming parts. ASME 1996; 3:227– 32. [5] Hillman M., Kubli W., Optimization of sheet metal forming processes using simulation programs. In: Numisheet 99, Besanc¸ France, 1999; 1:287–92.