FEM-based optimum design of multi-stage deep drawing process of molybdenum sheet

Journal of Materials Processing Technology 184 (2007) 354–362

FEM-based optimum design of multi-stage deep drawing process of molybdenum sheet Heung-Kyu Kim ∗ , Seok Kwan Hong Precision Molds and Dies Technology Team, Korea Institute of Industrial Technology, #401-301, Bucheon Techno-Park, 193, Yakdae-Dong, Wonmi-Gu, Bucheon, Kyunggi-Do 420-734, South Korea Received 28 November 2005; received in revised form 1 December 2006; accepted 1 December 2006

Abstract Molybdenum, one of the refractory metals, has high heat and electrical conductivity while remaining strong, mechanically, at high, as well as low, temperatures. Therefore, it is a technologically very useful material, especially for high temperature applications. However, due to its very low drawability, a multi-stage process is necessary to make a deep drawn part from the molybdenum sheet. In this study, a multi-stage circular cup deep drawing process for a molybdenum sheet was designed by including ironing, which was effective in increasing drawability. A parametric study by finite element analysis of deep drawing was conducted to evaluate the effect of die design variables. From parametric study results, the design variables of the multi-stage deep drawing process were selected. Then, nonlinear process optimization, based on finite element simulation, was conducted to obtain the optimum multi-stage deep drawing process, using a global, as well as a local, optimum search algorithm. © 2006 Elsevier B.V. All rights reserved. Keywords: Molybdenum; Ironing; Finite element analysis; Multi-stage deep drawing; Optimization

1. Introduction Molybdenum, with a melting point of about 2600 ◦ C, is mechanically strong at high, as well as low, temperatures [1–3] and thus can be used for high temperature applications. However, due to the low limit drawing ratio (LDR), a multi-stage process is necessary to make a deep drawn part from a molybdenum sheet. Manufacturing costs for a multi-stage process is related to the number of stages involved. Therefore, reducing the number of stages is one process design objective. For this reason, the respective stages should be carefully designed and arranged to make the desired part within the minimum possible number of stages without causing failure. Generally, design of such an optimum multi-stage process requires a tremendous amount of trial-and-error, using experimental or analytical approaches [4–6]. Thus, in many cases, one has to be satisfied with a good design rather than the optimum design. In the present study, circular cup deep drawing of a molybdenum sheet was investigated to evaluate the effect of die design

∗

Corresponding author. Tel.: +82 32 234 0606; fax: +82 32 234 0607. E-mail address: [email protected] (H.-K. Kim).

0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.12.001

variables using finite element (FE) analysis. Then, optimization, with respect to the die design variables, was conducted to find the most reliable multi-stage process using FE simulation, to evaluate the objective function value. Because the multi-stage deep drawing process is highly nonlinear, a global, as well as a local, search algorithm, was employed for optimization. Owing to the tremendous amount of calculation time involved, optimization was conducted under the constraint of a fixed number of stages, though the final objective of optimization was to design a process with the minimum number of stages. Instead, we regarded the obtained optimum design as a promising candidate to reduce the number of stages. 2. Plastic behavior modeling of molybdenum sheet Molybdenum sheet usually has a low LDR and therefore should go through many drawing stages to be transformed into a deep drawn cup. Accurate material properties of the molybdenum sheet should be evaluated to design the maximum stroke per stage and to provide the material data for FE analysis. For this reason, in the present study, a uniaxial tension test was conducted on molybdenum sheet to obtain the flow stress curve. As the as-received 99.97% molybdenum sheet has planar anisotropy, the test was conducted on specimens cut at three

H.-K. Kim, S.K. Hong / Journal of Materials Processing Technology 184 (2007) 354–362

Fig. 1. Stress–strain curves at 0◦ , 45◦ and 90◦ obtained by the tension test.

different orientations of 0◦ , 45◦ and 90◦ from the as-received sheet. The test results for the respective orientation specimens are shown in Fig. 1. The results show that the maximum strain, which corresponds to the strain just before specimen fracture, does not exceed ∼0.1 in all specimens. Though the molybdenum sheet has anisotropy in the sheet plane, as shown in Fig. 1, for simplicity, we assumed the sheet as an isotropic material because three-dimensional FE analysis, using the anisotropic material model, is computationally too expensive for the optimum design search in the present study. In addition, ironing, which is included in the multi-stage deep drawing process, makes the cup earing uniform compared with the pure drawing process, because the cup wall thickness reaches the specified clearance by the forced thinning action during ironing. In other words, the resulting cup deformation after the combined ironing–drawing multi-stage is more isotropic than that after the pure drawing multi-stage. Thus, we assume the isotropic material model is a good approximation in the present study. From the above, the flow stress curves for the three orientations were averaged, similar to r-value averaging, as: σ¯ =

σ0 + 2σ45 + σ90 4

355

From Eqs. (2) and (3), it can be assumed that the critical strain in the plane-strain drawing, ε¯ ∗ , is about 1.1547/2 times that in uniaxial tension, ε∗1 . Therefore, we can approximate 1.1547/2 times the limit strain in the uniaxial tension test as the limit strain in the single-stage cup drawing. This limit strain provides an important criterion for single-stage drawing process design. However, as discussed below, the multi-stage deep drawing process includes ironings, as well as pure drawings, and the failure behavior for ironing is different from that for drawing. In other word, the failure criterion discussed above cannot be applied to design the stage that includes ironing. In general, failure can be influenced by process conditions, such as lubrication, die material and material defects, as well as the material deformation and precise identification of an accurate failure criterion is beyond the scope of the present study. Therefore, a limit strain criterion was used to determine failure during each stage in the following FE analysis. 3. Initial multi-stage deep drawing process design and experimental trial Final target cup dimensions and the cup design variables for the intermediate stages are shown in Fig. 2 and Table 1. The initial blank diameter was 6 mm. Initially, an eight-stage process was designed, as shown in Table 2, based on the following rule of thumb: gradual increase in each die design variables (die corner radius, intake angle, etc.), volume constancy throughout the stages and an ironing process inclusion at every even-numbered stage (i.e. #2, #4 and #6), except #8 stage. To evaluate the initial design, multi-stage deep drawing was conducted using machined drawing dies. Owing to the low drawability of the molybdenum sheet, as well as the difficulty of

(1)

Thin sheet specimens, loaded in the uniaxial tension test, experienced localized necking, which finally leads to a failure of the specimen. If the flow stress behavior of the sheet is described as a power law, σ¯ = K¯εn , the critical strain for the localized necking becomes [7] ε∗1 = 2n

(2)

where the stretching direction is denoted as 1. However, critical strain for localized necking under plane-strain conditions is necessary to predict the failure in the drawing process, because the main deformation in cup drawing is plane-strain stretching instead of uniaxial tension. Thus, the critical effective strain can be calculated, similar to uniaxial tension, as 2 ε¯ ∗ = √ n ∼ = 1.1547n 3

(3)

Fig. 2. (a) Final target cup dimensions and (b) cup design variables for the intermediate stages. Table 1 Description of die design variables based on cup design L Di Do Rd Rp θ

Stroke Punch diameter Die diameter Die corner radius Punch corner radius Intake angle

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Table 2 Initial design of die design variables (clearance c = 0.5(Do − Di )) Stage

Do

c

L

Rp

Rd

θ (degree)

#1 #2 #3 #4 #5 #6 #7 #8

4.00 3.20 2.60 2.35 2.10 1.92 1.78 1.70

0.2 0.15 0.15 0.12 0.12 0.10 0.10 0.10

1.85 2.00 2.73 3.40 3.63 5.09 4.80 5.10

0.80 0.60 0.60 0.40 0.30 0.20 0.10 0.10

0.40 0.50 0.50 0.50 0.50 0.63 0.50 0.50

– 30 30 30 30 30 30 30

the deep-drawing process itself, in most cases, the experimentally obtained product showed failure in the wall, near the cup bottom, at stage #7. The experimental trial results are shown in Fig. 3. 4. Parametric study of design variables by FE simulation The ratio of wall thickness to cup diameter, in the present study, is relatively large compared with other drawn cups, which are assumed as sheet metal product. In addition, the thickness of the sheet is reduced by the ironing process. Therefore, solid element modeling is more suitable for FE analysis of the local material flow, caused by the ironing, than shell element modeling. Using a solid four-node element, the molybdenum sheet was modeled with multiple numbers of finite elements across the thickness direction. As discussed above, because ironing usually provides more uniform cup earing height and wall thickness distribution than pure drawing, and because three-dimensional FE analysis, considering the anisotropy of the material, is computationally expensive for multi-stage simulation, a two-dimensional axisymmetric FE analysis, based on isotropic material behavior, was conducted in the following parametric study and process optimization. For FE simulation, the commercial nonlinear FE analysis program, MSC.Marc, was utilized.

The wall thickness of the cup must be reduced from 0.2 to 0.1 mm during the multi-stage deep drawing because the thickness of the as-received sheet was 0.2 mm. The wall thickness reduction at each stage can be accomplished by controlling the clearance c, between the punch and the lower die. If the clearance is smaller than the wall thickness, the current stage is regarded as an ironing process. If not, the current stage is regarded as a drawing process. For simplicity, we consider only the cases when the clearance is smaller than or equal to the wall thickness. In this case, we have c = t for drawing and c < t for ironing, where t is the wall thickness. To evaluate the ironing effect on final cup quality, FE analysis was conducted for six cases, which have different sequential clearance designs. In the present study, for simplicity, only the clearances at stages #2, #3, #4 and #5 were varied as shown in Table 2. In FE analysis results for these cases, necking of the sheet was predicted in some cases because multiple numbers of finite elements were used to model the sheet across the sheet thickness direction. The necking predicted by FE analysis was regarded as a kind of failure because no further FE simulation was possible due to the excessive deformation of the elements. The maximum effective strain after the final stage, ε¯ #8 max , or the necking-occurring stage, predicted by FE analysis, are shown in Table 3. The friction coefficient used for FE analysis was 0.1. In the current molybdenum cup deep drawing, it can be inferred from FE analysis results that clearance selection at the early stages should be carefully determined. For example, stage #2 should be drawing rather than ironing because the maximum effective strain after the final stage was lower or the process was more successful in cases of drawing rather than ironing at stage #2. Considering material flow prediction by FE analysis, it can be inferred that the flange formed in the early stages hindered material flow during the later stages. Therefore, to improve material flow, the flange should be minimized as much as possible by drawing it as much as possible in the early stages. The maximum stroke at each stage, without causing failure, can be controlled by the clearance. For evaluating the maximum

Fig. 3. Experimental trial result at each stage for the initial design. Table 3 Different sequential clearance designs at stages #2, #3, #4 and #5, and FE analysis prediction (Case 5: initial design defined in Table 2) Stage

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

#2 #3 #4 #5

0.20, drawing 0.20, drawing 0.15, ironing 0.12, ironing

0.20, drawing 0.15, ironing 0.15, drawing 0.12, ironing

0.20, drawing 0.15, ironing 0.12, ironing 0.12, drawing

0.15, ironing 0.15, drawing 0.15, drawing 0.12, ironing

0.15, ironing 0.15, drawing 0.12, ironing 0.12, drawing

0.15, ironing 0.12, ironing 0.12, drawing 0.12, drawing

ε¯ #8 max

2.70

2.63

2.66

Necking at #4

2.66

Necking at #4

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Fig. 4. Predicted maximum stokes during stage #6 for die corner radius = 0.5. (a) Drawing, c = 0.12 and (b) ironing, c = 0.09.

stroke, FE analysis was conducted using different clearances for stage #6 of the initial design. Pushing the punch downward until necking occurred, the maximum strokes possible were predicted for drawing and ironing, as shown in Fig. 4. However, the maximum stroke was also affected by die corner radius, Rd .

Using a corner radius of 2.0 instead of 0.5, the maximum stroke decreased, as shown in Fig. 5. Die intake angle, θ, as well as die corner radius, had an effect on the maximum stroke. For evaluation of the effect, FE analysis was conducted by using different intake angles for stage #6 of

Fig. 5. Predicted maximum stokes during stage #6 for die corner radius = 2.0. (a) Drawing, c = 0.12 and (b) ironing, c = 0.09.

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Fig. 6. Predicted maximum stokes during stage #6 for ironing, c = 0.09. (a) θ = 15◦ , (b) θ = 30◦ and (c) θ = 60◦ .

the initial design. FE analysis shows that the maximum stroke increases as the intake angle increases, as shown in Fig. 6. The reason is because the frictional force between the flange and the die hindered material flow into the punch–die gap and the flange–die contact area decreases as the intake angle increases. 5. Optimum design of multi-stage deep drawing process based on FE simulation 5.1. Optimization procedure As discussed above, various die design variables have an effect on the feasibility of the desired deep drawing process. Therefore, the optimum multi-stage deep drawing process design requires consideration of the effects of those die design variables. However, the effects of the variables are inter-related due to nonlinearity in die geometry and material response during the multi-stage process. In this case, the optimum process design obtained for one die design variable may deviate from the optimum state if the other die design variables change. Therefore, the optimum design of the nonlinear process requires numerous iterative calculations considering the structure of the design space. For example, the design space may be as shown in Fig. 7, where there are many local minima, as well as one global minimum, of the objective function f(X). If the die design variable effects are considered individually for optimum design, the final optimum may be not the global optimum, but the local optimum. To avoid this, a global optimization algorithm, considering the nonlinearity of the problem, should be applied before the local optimization algorithm is conducted. In the present study, simulated annealing (SA) was applied as the global optimization algorithm, because SA is known to

be very well suited for solving highly nonlinear problems and, thus, effective in approaching the global optimum [8–10]. SA is based on the random evaluation of the objective function, in such a way that escape from a local minimum is possible. SA is effective for difficult problems with numerous local minima. The algorithm of SA can be described as follows [11]. The optimization problem is to find design X, which minimizes the objective function f(X) subject to XLB ≤ X ≤ XUB , where XLB is the lower and XUB is the upper boundary of the design variable. In addition, the following function is used during the optimization.   f P(f ) = exp − (4) kT In the above function, f is the objective function variation for design update, k a scaling factor, called the Boltzmann constant and T is the temperature. At the beginning of the optimization procedure, Boltzmann’s constant, k, initial temperature, T0 ,

Fig. 7. Local and global minimum of the objective function f(X).

H.-K. Kim, S.K. Hong / Journal of Materials Processing Technology 184 (2007) 354–362

359

algorithm of the HJ method can be described by the following steps [11]. Step 1. Set the starting base point, X1 , the step lengths, xi , and the tolerance for convergence check, δ. Step 2. Compute fk = f(Xk ). Set i = 1 and Yk,0 = Xk . Step 3. Exploratory move. A new temporary base point is obtained by perturbing about the current base point Yk,i−1 . Compute fk+ = f (Yk,i−1 + xi ui ), fk− = f (Yk,i−1 − xi ui ). For i = 1, 2, . . ., n, continue the following process. if fk < min(fk+ , fk− ),

Yk,i = Yk,i−1 , Fig. 8. Plot of the probability function, P, used for SA (Eq. (4)).

lower boundary temperature for convergence check, TLB , initial design guess, X* and tolerance for convergence check, δ, should be determined. Then, the optimum design XOPT is obtained after the following steps (Fig. 8). Step 1. Step 2. Step 3. Step 4. Step 5.

Step 6.

Step 7. Step 8. Step 9. Step 10.

Set T = T0 , T = 0, T ≤ TLB , k, n = 0, X* , δ. Set T = T − T. Set X0 = X* , i = 0. Generate random move, X, based on random number generation, and update design by Xi+1 = Xi + X. Accept or reject the updated design by the following criterion. Calculate f = f(Xi+1 ) − (Xi ). If f ≤ 0 accept the new point as candidate design X* = Xi+1 , else f > 0 accept the new point with a probability of P(f) = exp(−f/(kT)). Check if the thermal equilibrium is reached. If |f| ≤ δ for the small tolerance, δ, and during the pre-determined iteration number, then go to Step 8. Set i = i + 1 and go to Step 4. If T ≤ TLB , then go to Step 10. Generate temperature reduction T and go to Step 2. Set optimum design XOPT = X* .

However, though the global optimum can be found by controlling the parameters in SA for precise searching, it may take too long because SA is more effective for a global search than a minute local search. Therefore, it is computationally more efficient to terminate SA after obtaining a design near the global optimum and, then, to apply another optimization method, which is effective for a local optimum search. In the present study, the Hooke–Jeeves (HJ) method is used for this purpose. The HJ method consists of two types of moves: an exploratory move and a pattern move. The exploratory move explores the local behavior of the objective function, while the pattern move takes advantage of the pattern direction, which is obtained from the exploratory move. The HJ method is expected to find the real optimum design precisely around the optimum design obtained by SA because it is effective in exploring the local behavior of the objective function. The

Yk,i = Yk,i−1 + xi ui ,

if fk+ < min(fk , fk− )

Yk,i = Yk,i−1 − xi ui ,

if fk− < min(fk , fk+ )

Step 4. If Yk,n = Xk , reduce the step length xi , set i = 1, and go to Step 3. If Yk,n = Xk , and go to Step 5. Step 5. If f(Yk,n ) < f(Xk ), set the new base point as Xk+1 = Yk,n and go to Step 6. If f(Yk,n ) ≥ f(Xk ), set Xk+1 = Xk , reduce the step length xi , set k = k + 1, and go to Step 2. Step 6. Pattern move. Establish a pattern direction S as S = Xk+1 − Xk , and find a point Yk+1,0 = Xk+1 + λS, where the step length, λ, is determined by solving a 1D minimization problem in the direction S. Step 7. Set k = k + 1, fk = f(Yk,0 ), i = 1, and go to Step 3. Step 8. Convergence check. Whenever max(xi ) < δ during the above steps, teri

minate the process and set XOPT = Yk,n . Of course, the finally obtained design still may not be the global optimum if the optimization algorithm and the used parameter values are not appropriate to search the whole design space of the current problem thoroughly. In the present study, however, we regard the design obtained by applying SA and then HJ as the optimum design sought because the obtained design is better than the initial design, at least, for the considered design variables and objective function. For the optimization based on the above algorithms, FE simulation should be conducted to provide the objective function values for the temporary designs. Because the objective function must be evaluated at many steps for one cycle of the optimization procedure, numerous FE simulation of the whole multi-stage process must be conducted during the entire optimization procedure. In this case, even starting the FE program for the generated design becomes a tremendously time-consuming task. To overcome this difficulty, we utilized the commercial optimization program iSIGHT, which automatically carried out the selected optimization procedure, as well as running the FE program for the generated design by communicating with the FE program

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H.-K. Kim, S.K. Hong / Journal of Materials Processing Technology 184 (2007) 354–362 Table 5 Design range for punch diameter ∅ Stage

Lower bound

Initial value

Upper bound

#2 #3 #4 #5 #6 #7 #8

1.5 1.5 1.5 1.5 1.5 1.5 –

3.4 2.8 2.3 1.9 1.5 1.5 1.5

3.4 3.4 3.4 3.4 3.4 3.4 –

Table 6 Design range for clearance Stage

Lower bound

Initial value

Upper bound

#2 #3 #4 #5 #6 #7 #8

0.1 0.1 0.1 0.1 0.1 0.1 –

0.15 0.15 0.12 0.12 0.09 0.1 0.1

0.2 0.2 0.2 0.2 0.2 0.2 –

Fig. 9. Schematic diagram of the optimization procedure.

through the macroinput/output file [12]. That is to say, iSIGHT conducted, by itself, the task within the dotted block in Fig. 9, which shows a schematic diagram of the overall optimization procedure. To conduct an actual optimization, the objective function, design variables and constraints must be determined. From the above parametric study results, the punch diameter, clearance and stroke, at each stage, were selected as the design variables for optimization, while the punch corner radius, die corner radius and intake angle were excluded from the design variables and given fixed values to save computational costs, as shown in Table 4. Therefore, the number of design variables becomes 24 for the eight-stage process. Because failure was closely related to excessive strain, we assumed the objective function as: f (xi ) = W1 × ε¯ final max + W2 ×

1

(5)

final tcup bottom

where ε¯ final max is the maximum effective strain after the final stage final and tcup bottom is the cup bottom thickness after the final stage, which was introduced to satisfy the target cup bottom thickness of 0.2 mm. Though the target cup bottom thickness can be Table 4 Design and fixed variables for an eight-stage process Stage

Do

c

L

Rp

Rd

θ (degree)

#1 #2 #3 #4 #5 #6 #7 #8

x.xx x.xx x.xx x.xx x.xx x.xx x.xx x.xx

x.xx x.xx x.xx x.xx x.xx x.xx x.xx x.xx

x.xx x.xx x.xx x.xx x.xx x.xx x.xx x.xx

0.80 0.60 0.60 0.40 0.30 0.20 0.10 0.10

0.40 0.50 0.50 0.50 0.50 0.63 0.50 0.50

– 30 30 30 30 30 30 30

satisfied only if no material sheet flow occurs over the punch bottom, optimization is expected to give the final cup shape, in which the bottom thickness is near 0.2 mm, by including the term in the objective function. However, minimizing the maximum effective strain in the final cup is more important than cup bottom thickness for successful deep drawing without failure. Therefore, we used the larger weight factor, W1 = 10.0, for the maximum effective strain term than the weight factor, W2 = 1.0, for the cup bottom thickness term. Considering the feasible range of design variables, the upper and lower bounds of the variables were selected arbitrarily, as shown in Tables 5–7. If the maximum effective strain exceeds its critical value, or the cup bottom thickness becomes smaller than its critical value, during the intermediate stage, no further FE simulation of the multi-stage process is necessary because the designed process is unsuccessful. Then, FE simulation should be terminated at this stage and resumed from the initial stage for the new design generated by the optimization algorithm. The whole calculation flow along the multi-stage is shown in Fig. 10. We used 2.0 for the critical value of the maximum effective strain and 0.1 mm for the critical value of the cup bottom thickness. Table 7 Design range for stroke Stage

Lower bound

Initial value

Upper bound

#2 #3 #4 #5 #6 #7 #8

1.6 1.6 1.6 1.6 1.6 1.6 –

2.2 2.6 1.9 2.4 3.9 5.0 5.0

5.0 5.0 5.0 5.0 5.0 5.0 –

H.-K. Kim, S.K. Hong / Journal of Materials Processing Technology 184 (2007) 354–362

361

Fig. 10. Calculated flow along the multi-stage for a generated design.

In the actual FE simulation, as shown in the parametric study, sheet necking was predicted. Therefore, necking of the sheet, as predicted by FE simulation, was utilized as an additional failure indicator, though the predicted necking may be dependent on FE mesh performance.

Table 8 Die design variables of the optimum design Stage

#1

#2

#3

#4

#5

#6

#7

#8

L Do c

1.85 2.3 0.2

0.423 1.744 0.1

2.686 1.637 0.2

2.798 1.588 0.198

2.726 1.5 0.120

1.849 1.328 0.133

2.321 1.311 0.137

5.01 1.05 0.1

5.2. Optimization result After iterative calculations, an optimum design of the multistage deep drawing process was obtained. The die design variables at each stage of the optimum design are summarized in Table 8, and the corresponding cup deformations by FE simulation are shown in Fig. 11. For comparison, the predicted maximum effective strains at each stage for the optimum design are shown in Table 9 compared with those for the initial design, which is shown in Table 2. A comparison of results shows that the maximum effective strains for the optimum design (B) is only 60–80% of that for the initial design (A) at the respective stages. This suggests that the cups for the optimum design are

Table 9 Predicted maximum effective strains, ε¯ max , at each stage for the initial and optimum design Stage

Initial design (A)

Optimum design (B)

Strain ratio (B/A)

#2 #3 #4 #5 #6 #7 #8

1.11 1.59 1.86 2.12 2.56 2.66 2.78

0.77 1.32 1.43 1.54 1.53 1.55 2.16

0.69 0.83 0.77 0.73 0.60 0.58 0.78

Fig. 11. FE-simulated cup deformations for the optimum design.

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free from failure, while the cups for the initial design are not, if failure occurs at any strain between A and B. In other words, the optimum design is safer than the initial design for failure. In addition, the obtained optimum design is an initial candidate for reducing the number of stages in the design, though this was not attempted in the present study. 6. Conclusion In the present study, the optimum design of the multi-stage deep drawing of a molybdenum sheet was conducted by FE simulation-based optimization. First, a parametric study of die design variables was conducted to evaluate their effect on failure characteristic, using nonlinear FE analysis of the deep drawing process. From the parametric results, it could be inferred how design variables, such as clearance, die corner radius and die intake angle, affected maximum possible stroke per stage, material flow and cup quality. However, owing to the severe nonlinearity of the multi-stage deep drawing process, an effective nonlinear optimization algorithm was necessary to obtain the optimum design. Therefore, optimization, based on a global, as well as a local search algorithm, was applied to find the optimum design, which actually gave cup deformations with lower possibility of failure after the final, as well as the intermediate stage, than the initial design, in terms of the maximum effective strain prediction. An iterative nonlinear optimization procedure was conducted automatically after inputting the selected optimization method (SA and HJ) and the required values for the die design variables into the commercial optimization program, iSIGHT. The study showed the possibility of FE analysis-based optimization in a highly nonlinear process, such as the multi-stage deep drawing process. The optimization procedure used here may be applied to other nonlinear processes, which, generally, have numerous local optima and require FE analysis for singlestage simulation. In future, the failure mechanism during ironing should be investigated to accurately apply the failure criterion to FE analysis-based optimization. In addition, three-dimensional FE

analysis considering the anisotropy of the molybdenum sheet should be conducted for more precise process optimization. Acknowledgements The authors acknowledge the financial support from the Korea Ministry of Commerce, Industry and Energy and the InterResearch Consortium for Materials & Components Technology. The authors also gratefully acknowledge the technical support by Hahntack Lee of Engineous Korea Inc. References [1] S. Nemat-Nasser, W. Guo, M. Liu, Experimenally-based micromechanical modeling of dynamic response of molybdenum, Scripta Mater. 40 (7) (1999) 859–872. [2] J. Cheng, S. Nemat-Nasser, W. Guo, A unified constitutive model for strainrate and temperature dependent behavior of molybdenum, Mech. Mater. 33 (2001) 603–616. [3] L. Hollang, D. Brunner, A. Seeger, Work hardening and flow stress of ultrapure molybdenum single crystals, Mater. Sci. Eng. A319–A321 (2001) 233–236. [4] D.K. Min, B.H. Jeon, H.J. Kim, N. Kim, A study on process improvements of multi-stage deep-drawing by the finite-element method, J. Mater. Process. Technol. 54 (1995) 230–238. [5] S.H. Kim, S.H. Kim, H. Huh, Tool design in a multi-stage drawing and ironing process of a rectangular cup with a large aspect ratio using finite element analysis, Int. J. Mach. Tool Manuf. 42 (2002) 863–875. [6] T.W. Ku, B.K. Ha, W.J. Song, B.S. Kang, S.M. Hwang, Finite element analysis of multi-stage deep drawing process for high-precision rectangular case with extreme aspect ratio, J. Mater. Process. Technol. 130–131 (2002) 128–134. [7] W.F. Hosford, R.M. Caddell, Metal Forming, second ed., Prentice-Hall International, London, 1993. [8] S. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671–680. [9] L. Ingber, Simulated annealing: practice versus theory, Math. Comput. Model. 18 (1993) 29–57. [10] Z.G. Wang, M. Rahman, Y.S. Wong, J. Sun, Optimization of multi-pass milling using parallel genetic algorithm and parallel genetic simulated annealing, Int. J. Mach. Tool Manuf. 45 (2005) 1726–1734. [11] S.S. Rao, Engineering Optimization, Theory and Practice, third ed., Wiley Interscience, New York, 1996. [12] iSIGHT Reference Guide, Version 9.0, Engineous Software, USA, 2004.