the blank material, so as to  avoid  the  defects of fracture, wrinkle and serious thinning。

2。3Finite element modeling

In this work, we use DYNAFORM to simulate the deep drawing process of the head, and establish finite element modeling based on the actual die structure, as shown in Fig。 3。 Numerical simulation parameters are set as follows: the material of head is BTC 330R; the elastic modulus is 289 GPa; the Poisson ratio is 0。28; the hardening  coefficient  is  568。9  MPa;  the  hardenability

value  is  0。34;  and  the  anisotropic  parameter  is 1。488;

Fig。 4 Schematic diagram of objective functions definition

diameter of punch (Dp) is 343。6 mm, and diameter of die (Dd) is 350 mm。 Meanwhile, punch, die and blank holder default as being rigid。 The friction coefficient of sheet metal contacting with rigid die is 0。12。

The objective function Dt is defined as

where k represents the number of units which is tested; t0 represents the initial sheet thickness; ti represents  the final sheet thickness。

Fig。 3 Finite element model of drawing: 1−Die; 2−Blank; 3−Binder; 4−Punch

3Multi-objective optimization based on response surface method

The best combination of process or geometry design variables can be got using RSM, which will lead to a desired sheet metal part without any defects, such as fracture, wrinkles and thickness varying, etc。

3。1Objective function

The forming defects can be quantified by objective function, which can judge the formability of sheet metal parts as criteria。 In this work, the fracture, wrinkle and thickness varying are considered the optimization targets。 According to the definition of wrinkle and fracture forming limit curves raised in Ref。 [6], the fracture Df, wrinkling Dw, and thickness varying Dt corresponding objective function are constructed as follows, based on the constraints of forming limit curve, as shown in Fig。 4。

The fracture function Df can be formulated as

3。2

Experimental design

In most cases, the relationship between the response variables and the independent variables are unknown。 In the present experimental investigation, response surface methodology is applied to establishing polynomial equation so as to get the optimal combination of parameters, in terms of the efficiency and accuracy of optimization。 Finding a suitable approximation  to  the true relationship is the first step in RSM。 Meanwhile, determining reasonable variables and levels is significant。

Thus, we adopt the same idea as ZHOU et al [2]—central

composite design and steepest ascent design  were adopted to arrange variables and experiments。 There are four variables, i。e。 the fillet radius, the position of draw-bead, the blank size and the blank-holding force。 The four variables were respectively represented by x1, x2, x3, x4, and the fracture function, the wrinkling function and the thickness function are respectively   represented by Df, Dw and Dt for brevity。

The steepest ascent design is introduced in the present investigation for its efficiency。 With this technique, the region for each variable is correctly defined。 The results of steepest ascent design are listed in Table1。

It  shows  that  the  minimum  values  of  the    three

objective values appear when the experiment is tested  at

the parameter combination of No。3。 It can be   concluded

The wrinkling function Dw can be formulated as

that the optimum parameter combination will exist   near

the parameter combination of No。3。 Thus, the center point of response surface experiments is set at (3,    385,