September 1 - 5, 2008 – Interlaken, Switzerland

Sheet MetaL hYDrOMeChaNICaL Deep DraWING prOCeSS OptIMIZatION antonio Del prete1*, Barbara Manisi1, Matteo Strano2 1

University of Salento, Department of Innovation Engineering University of Cassino, Department of Industrial Engineering

2

aBStraCt: Sheet metal panel hydroforming is a well known technique, recognized as an alternative

for medium-low volume productions. The technological opportunity and potential of the process is often not fully exploited due to the difficulty in managing the large number of process variables: fluid pressure, blank holder force, preforming height, etc. Experimental equipment is described in the paper, whose most important peculiarity is the actuation of the blankholder ring, compressed by the coordinated action of 12 hydraulic actuators, which can be independently controlled vs. time. An optimization technique has been developed, in order to optimize the process response as a function of multiple blankholder forces and fluid pressure curves vs. time. The proposed procedure is based on a metamodel of the simulation response by means of Kriging interpolation. Application of the optimization technique to a reference test case is demonstrated.

KeYWOrDS: sheet hydroforming, optimization, CAE 1

INtrODUCtION

response as a function of the blankholder forces and fluid pressure curves. The proposed procedure (Section 4) is based on the creation of a metamodel of the simulation response by means of Kriging interpolation [4]. In order to cope with the potentially very large dimensionality of the design problem, a simplifying method has been used, for the indirect determination of the multiple blankholder force vs. time curves. In Section 5, application of the optimization technique to a reference test case (previously introduced in Section 3) is demonstrated.

Sheet metal panel hydroforming is a well known technique. It has been recognized as a real opportunity for medium-low volume productions in alternative to conventional metal forming processes. The technological opportunity and potential of the process is often not fully exploited due to the difficulty in managing the high number of process variables: fluid chamber pressure, blankholder force distribution, preforming height, etc. In the next Section 2, the equipment used for the present study is described, which has been designed and assembled as part of a larger research program [1]. The most important peculiarity of this machine is the actuation of the blankholder ring, which is compressed by the coordinated action of 12 hydraulic actuators, which can be independently controlled vs. time. Designing an hydroforming operation for this press is a difficult and complex task, due to the presence of multiple loading curves vs. time. A numerical simulation model (based on Ls-Dyna as solver) has been previously tuned to the experimental results obtained with this sheet hydroforming cell, and integrated into a specifically developed software tool, called Hydroforming Manager [7]. Starting from this model, a novel optimization technique has been developed, in order to optimize the process

2

eXperIMeNtaL eQUIpMeNt

The sheet hydroforming cell used in the experiments has been designed thanks to output data (maximum punch stroke, maximum preforming height, maximum required volume for the fluid chamber, blank shape, blankholder dimensions, magnitude of process variables, etc) obtained from CAE analysis [3]. Structural analysis on the cell components has allowed to determine the best solution for typology and number of actuators. A blankholder solution which is able to produce a holding force variable during the process time and along the tool profile has been implemented. In order to realize this kind of equipment, a solution with twelve actuators has been chosen (Figure 1). The thickness of the upper

____________________ * Corresponding author: via per Arnesano Complesso Ecotekne, 73100 Lecce, Italy, phone/fax: +390832297809, email: [email protected]

685

Numisheet 2008

September 1 - 5, 2008 – Interlaken, Switzerland

blankholder plate has been investigated since it has an important influence on the actual transmitted holding force to the blank during the forming process.

Each of them receives signals by a proportional valve and is protected from overload pressure by a maximum pressure valve. Actuators apply their load on the upper blankholder.

3

teSt CaSe

The punch shape shown in Figure 3 (where a ¼ model is shown) has been used as a test case in this study. The punch dimensions are 450x350x150 mm and the reverse drawing height is equal to 20 mm. All the fillet radius are 10 mm.

PUNCH

Figure 1: detail of actuators and upper blankholder plate. BH2

BH3

In the tooling design and development phase, a modular approach has been chosen to obtain reduced costs and time cycle for tooling substitution.

BH1

BLANK DIE

Figure 3: Test case.

Due to the double symmetry of the part, 3 independent actuators must be controlled (instead of 12). The portions of blankholder where they operate are shown in Figure 3 and labelled BH1, BH2, and BH3. Hence, the design variables of this process are the three forces BHFi(t) vs. time, the fluid pressure P(t) vs. time, the total drawing depth dd, the total punch stroke s and the maximum possible initial prebulging height h=s-dd. If fixing the drawing depth dd=100 mm, the prebulging height h=45 mm and, hence, the total stroke s=145 mm, the feasible process window appears to be very small for this specific operation. As a matter of fact, several attempts of designing this process on a FeP04 steel with initial thickness t0=0.7 mm have failed to provide a feasible part: in Figure 4 the failure risk simulated by the FEM and the actual part are shown.

Figure 2: Hydroforming cell.

The hydroforming cell (Figure 2) is made up of two parts: the lower part is made by the fluid chamber, positioned onto a support plate (lower base) [2]. Over the fluid chamber is positioned the lower blankholder ring. The upper part is connected to the press table. It is composed by a support plate, named upper base, where the active components of the cell are mounted: punch and twelve hydraulic actuators. The punch is connected to a structure, the punch–holder base connected to the upper base. This structure is designed to support each punch thanks to proper adapters. Within the framework of the upper base, actuators are connected to the hydraulic system of the cell.

Figure 4: Part with h=45 mm; real process fails when stroke s reaches 80 mm and dd=35 mm, FE model.

686

Numisheet 2008

September 1 - 5, 2008 – Interlaken, Switzerland

4

OptIMIZatION MethOD

For the formulation of the problem presented in the previous Section 3, an objective function has been chosen as quality indicator y, dependent on the minimum tmin and maximum tmax values of thickness registered after each simulation run, measured within a bounding box which excluded the out of figure part. 2

Figure 5: Part with h=45 mm; real process fails when stroke s reaches 80 mm and dd=35 mm, real model.

§ tmax · §t · 1.05 ¸ ¨ min 0.95 ¸ ¨ © t0 ¹ © t0 ¹

y

2

(1)

As far as the design variables are concerned, first of all, the fluid pressure vs. time curve P(t) must be considered. If assuming that the final value, i.e. the calibration pressure, can be calculated in advance (15 MPa in this example), the rest of the curve must necessarily be segmented in a finite number of values in order to solve the problem. As an example, three points could be identified vs. the punch stroke s (Figure 8): the prebulging pressure P1, when the stroke reaches the level of the die (45 mm), the pressure P2, when the stroke reaches about half of the drawing depth and the pressure P3, when the stroke is nearly finished.

Indeed, the experimental evidence has proved that the prebulging height (h=45 mm) is too large: if decreasing the prebulging height down to h=15 mm, drawing depths up to dd=150 mm can be obtained with conventional trial-and error process plans: in Figure 6 and 7, a safe part is shown along with its FEM simulation.

15

P3

[MPa]

12 9 6

P1

3

Figure 6: Safe part with h=15 mm; dd=150 mm, FE model.

P2

pressure vs. stroke

0 0

50

100

150

[mm]

200

Figure 8: Example of pressure vs. stroke curve

Similarly, the three independent BHF vs. time curves could be represented by three keypoints each. However, this would yield a total of 12 design variables to be optimized. This would require, even for a preliminary scan made by a simple fractional 2N plan, not less than 2000 runs, which is abundantly over realistic industrial computational times. For this reason, an indirect method of determining the various BHF vs. time curves has been developed, described in the following subsection.

Figure 7: Safe part with h=15 mm; dd=150 mm, real model.

In order to test and verify the effectiveness of the proposed optimization method, it has been tested on the operation with supposedly small process window (i.e. h=45 mm).

4.1 DeterMINatION OF BhF CUrVeS In the FEM simulations run with Ls-Dyna, the vertical position of the blankholder is not controlled by applied force or pressure loads, but it is kept fixed, allowing a predetermined gap, which is less than 0.1 mm greater than t0. Thickening and wrinkling of the blank under the flange is still 687

Numisheet 2008

September 1 - 5, 2008 – Interlaken, Switzerland

allowed because the tools are rigid bodies modelled with shell elements with a given thickness. Contact reactions develop as elastic forces as the blank and the shells of the tools penetrate each other. The actual level of restraint imposed by the BHi can be varied by regulating the contact penalty constants used by Ls-Dyna to determine the reaction forces (Figure 9) [5]. This can be done by scale factors wi which are assigned to the interface contact definitions of each BHi. Greater values of scale factor produce larger elastic stiffness of the contact and, therefore, produce larger blankholder force. As a consequence, each BHFi vs. time curve can now be produced by the simulation as an output response, and not as an input loading curve. The three scale factors wi can now be used as design optimization variables, together with P1, P2 and P3. In calculations, the input design variables are handled as a vector x made by 6 components: x ª¬ P1 P 2 P3 w1

w2

w3 º¼

simulation required for building the model is (N+1)(N+2)=28. Therefore an initial batch of at least 28 simulations must be selected in order to have a starting data set for the optimization. This is performed by Latin Hypercube random generation of 28 values of x, bounded within the following constraints: ° x min ® °¯ x max

gap

ª Pb ¬

0 0 0 0 [email protected] Pf

Pf

w1max

w2max

w3max º¼

(3)

where Pb is the maximum allowed pre-bulging pressure (2 MPa), Pf is the maximum allowed forming pressure (15 MPa) and wmax is equal to 0.24. After the initial batch of simulations has been generated and run, the metamodel can be built and optimized (minimized). Minimization is performed through a local constrained optimization method, based on a numerically calculated gradient approach. Since the optimization routine is local, it is run three times with different starting points in order to reduce the risk of finding local suboptimal solutions. If the quality of the initial optimization is not satisfactory, a second batch of additional 28 simulations can be generated, the metamodel rebuilt over the total 56 runs and constrained optimization performed again. These steps could be repeated a number of times. Clearly, if no satisfactory solution is found after a given number of iterations, the process can be declared as nonfeasible. If a feasible or satisfactory solution is found, a new FEM simulation can be run with the optimal values and the reaction forces BHFi(t) can be extracted by post-processing the simulation (considering that ¼ symmetry is used and, therefore, values of BHF1 and BHF3 must be doubled).

(2)

blankholder premilamierasector

(costante) (constant) constant)

>0

Spessore t 0 t0 Initial blanklamiera thickness

die surface matrice

Figure 9: Constant gap between die and BH plate.

The proposed approach presents some drawbacks and limitations, if compared to direct optimization of the BHF curves, but it allows considerable savings in the computational costs. The two main drawbacks are listed as follows. The quality of the solution depends on the predetermined gap: if it is too small, required BHF forces might be overestimated and no feasible solution might be found to tearing under the flange; if it is too large, contact might be lost and forces underestimated and no feasible solution might be found to wrinkling under the flange. The shape of the produced BHF(t) curves becomes correlated to the shape of the P(t) curve, since contact forces will increase as pressure increases; this might sound reasonable from a technological point of view, but it is still a limitation to the space where the optimal solution can be searched for.

5

appLICatION OF the prOpOSeD MethOD

The method described in Section 4 has been applied to the test case described in Section 3. An initial batch made of 28 runs has been conducted with Pb=2 MPa, Pf=14 MPa, w1max=0.25, w2max=0.12, w3max=0.20 and a blankholder gap which is 9 % greater than the initial wall thickness t0=0.7 mm. The matrix made of the 28 values of x is given in Table 1. The outcome of each simulation is synthetically expressed by the thickness uniformity indicator y, given in (1). The 6 runs with the minimum values of y are described with the aid of Figures 10 and 11, where the Pvalues and the w-values are reported. The run numbers are printed in the legends, in ascending order of y-value, i.e. the run number 24 is the best of this first batch. Figure 10 shows that 2 of the best simulations (runs 24 and 28) are obtained with

4.2 MetaMODeLING aND OptIMIZatION The technique of choice for metamodeling the objective function y (1) is the Kriging interpolation with a deterministic component based on a second order polynomial regression model [6]. For this kind of model, if N is the size of x (N=6 in the present example), the minimum number of 688

Numisheet 2008

September 1 - 5, 2008 – Interlaken, Switzerland

non monotonously increasing values of pressure and 2 more runs (no. 1 and 26) are obtained with linearly increasing values of P. In Figure 11 the values of w2 are smaller than the values of w3, which are smaller than the values of w1, but this is obviously due to the imposed boundaries w1max, w2max and w3max.

0.25 0.20 0.15 0.10 0.05

Table 1: matrix if the input values x run

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

[MPa]

16 14 12 10 8 6 4 2 0

P1

0.211 0.945 1.901 1.535 1.635 0.126 0.747 0.001 1.773 1.460 1.115 0.557 1.423 0.338 0.441 1.243 0.710 1.851 0.815 1.064 1.647 1.954 1.344 0.215 1.153 0.872 0.586 0.400

P2

3.689 7.430 3.427 0.297 4.748 13.450 10.911 1.823 1.409 6.330 5.171 9.645 13.671 8.767 10.364 6.655 9.419 12.284 11.185 8.351 12.773 5.722 2.847 11.811 2.397 4.478 0.761 7.940

P3

7.392 10.840 11.231 6.284 9.397 5.530 12.801 8.525 2.208 11.653 2.937 0.508 6.884 13.485 4.495 7.894 5.219 4.749 12.107 3.322 10.160 9.980 1.863 3.706 13.872 8.134 0.220 1.467

w1

0.212 0.104 0.090 0.218 0.147 0.021 0.010 0.167 0.140 0.060 0.049 0.068 0.073 0.236 0.042 0.115 0.083 0.172 0.195 0.225 0.007 0.124 0.199 0.154 0.242 0.126 0.184 0.029

w2

0.035 0.051 0.062 0.021 0.097 0.069 0.082 0.088 0.002 0.104 0.054 0.111 0.042 0.107 0.028 0.073 0.046 0.089 0.067 0.079 0.032 0.006 0.010 0.094 0.019 0.039 0.014 0.057

0.00

w3

w1

0.141 0.101 0.037 0.178 0.060 0.115 0.130 0.148 0.089 0.016 0.156 0.153 0.074 0.105 0.166 0.057 0.173 0.026 0.136 0.078 0.009 0.004 0.028 0.121 0.065 0.049 0.043 0.097

p2

w2

w3

Figure 11: Values of contact penalty constants for the best 6 simulation runs.

Once the 28 input values of x and output values of y have been obtained, the kriging metamodel function yˆ x of the response can be built and optimised, thanks to the constrained optimization method described in Section 3 and an optimal value xopt is obtained. At this point of the procedure, the user is given two choices: a simulation can be run with the xopt optimal input combination. If the result of this simulation is considered satisfactory, it can be further improved or refined, e.g. by changing the value of the calibration pressure (see the last point in Figure 8) or by changing the value of the initial gap. If the simulation out of the initial batch is satisfactory, the procedure can be repeated, i.e. another batch can be prepared and launched. If starting a new batch, the extreme values given in equation (2) could (should) be changed by the user, according the results of the first batch. At the end of the second batch, the metamodel yˆ x is built over 56 runs and optimized; the

24 25 26 28 27 1

p1

24 25 26 28 27 1

0.30

resulting simulation run with the new xopt can be improved, as described above. If the values of calibration pressure and blankholder gap are changed as well, the new metamodel must be built only on the last 28 runs. In the proposed example, none of the first 28 runs showed any fracture, but all simulations showed some degree of wrinkling. The optimal solution xopt of these first 28 runs generates the part shown in Figure 12, where some wrinkling is present. If refining this solution by increasing the calibration pressure from 15 to 20 MPa and reducing the blankholder gap from 9% to 0.1% of the initial thickness t0, the result shown in Figure 13 can be obtained, i.e. a part without fracture, no excessive thinning or thickening, correct coining of die radii, although some wrinkling remains at the corner of the pan. In order to obtain the loading curves for the hydroforming press, the output contact forces of the three blankholder segments can be extracted from the post processor. The result is shown in Figure 14.

p3

Figure 10: Values of pressure curves for the best 6 simulation runs.

689

Numisheet 2008

September 1 - 5, 2008 – Interlaken, Switzerland with a conventional trial-and-error design method proved very difficult to be found. The proposed method was able to find a partly acceptable solution (although with some wrinkling at the die corners) already within the first batch of simulations.

7

Authors are very grateful to “MUR: Ministero dell’Università e della Ricerca” for funding this work recognized as I.T.Idro: innovative solutions for sheet hydroforming. Special thanks are addressed to Stamec srl which has the role of industrial partner of the project.

Figure 12: result of the optimal simulation.

8

BHF3 BHF2 BHF1

[kN]

200 150 100 50 0 0

50 100 150 fluid pressure [bar]

200

Figure 14: optimal blankholder force vs. pressure curves out of the first 28 runs.

If this initial optimal solution is not considered satisfactory, a new batch of simulations can be run with a reduced gap width and an increased calibration pressure.

6

reFereNCeS

[1] Del Prete A., Anglani A., Papadia G.: Experimental Campaign Definition For Sheet Hydroforming Through CAE Tools Usage. NUMIFORM 2007. [2] Del Prete A., Papadia G., Manisi B.: Multi Shape Sheet Hydroforming Tooling Design. In 12° International Conference on Sheet Metal, SHEMET07, Palermo. [3] A. Del Prete, A. Anglani, T. Primo, A. Spagnolo: Non-Conventional Metal Forming Tooling set up through Computer Aided Simulation. In APCOM’07 in conjunction with EPMESC XI, 2007, Kyoto, JAPAN [4] Rijpkema J. J. M., Etman L. F. P., Schoofs A. J. G.: Use of Design Sensitivity Information in Response Surface and Kriging Metamodels. Optimization and Engineering, 2, 469–484, 2001. [5] Hallquist J.O.: LS-DYNA Theoretical manual. LSTC, 1998. [6] Lophaven S. N., Nielsen H. B., Søndergaard J.; DACE, A Matlab Kriging Toolbox, Technical Report of IMM at the Technical University of Denmark, IMM-TR-2002-13, 2002. [7] A. Del Prete, A. Elia, T. Primo, B. Manisi: Process Automation Tools Development for sheet Metal Hydroforming Simulation. In ISC’07, 5th International Simulation Conference, Delft, The Netherlands.

Figure 13: result of the refined optimal simulation

250

aCKNOWLeDGeMeNt

CONCLUSIONS

A method has been presented for the optimal design of a sheet hydroforming process with multiple loading curves vs. time. The method is based on the optimization of a kriging metamodel, built over the results of sequential batches of simulation runs. The method has been applied to a rectangular panel with both convex and concave radii on the bottom surface. Finding the correct process solution for the hydroforming of this panel (with given pre-bulging height and drawing depth), 690