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Fig.4 shows the flowchart of the BHF optimization strategy. The BHF is adjusted by PID controller at each optimization step so that the BHF is just large enough to avoid wrinkling. The optimization starts with an initial BHF. Then, wrinkling height (HW) will be calculated for each optimization step of forming simulation. If this wrinkling height is under the safe range of wrinkling (HW, S), then the current iterative BHF is the optimal BHF for the calculating step and will be treated as the initial BHF for the next optimization step. Otherwise, the wrinkling PID will be triggered to regulate the BHF so as to control wrinkling height under desired amplitude. The formula of wrinkling height PID controller is
Where kp, ki and kd are proportional, integral and differential gains of the wrinkling height PID controller;and e is the error between desired amplitude and current amplitude. They are updated at each iterative step as follows.
Once the wrinkling is under control, the strategy will query the cracking index. If crack occurs, it tells the BHF that could suppress the wrinkling also causes the cracking. This indicates no formable BHF window of
Fig.4 Flowchart of closed-loop PID control logic for VBHF optimization
Fig.3(c) and the optimization stops. If no crack occurs, the optimization continues to calculate the next optimization step until the step reaches the termination. With this strategy, the time-variable VBHF trajectory for the whole punch stroke could be output for each segmented binder in a single round of the forming simulation.
For implementing this strategy, the important operation is to enable the plot of the optimization step and restart analysis. To get around this problem, the “D3PLOT” and ‘‘D3DUMP”[16] output keywords have been implemented in the LS-Dyna input deck of the stamping simulation according to the optimization step set-up. At each optimization step, wrinkling height and cracking could be detected in D3PLOTnn plot and the BHF could be regulated according to the index level. Then, the input deck with updated BHF trajectory and D3DUMPnn could be submitted to LS-Dyna code for continuing calculation after the optimization step. With this implementation, the interaction between the strategy and the FEM code could be realized.
3 Determination of optimal VBHF trajectories for Numisheet’05 deck lid
3.1 Design of segmented binders