Time-varying process control for stringer sheet forming by a deterministic derivative-free optimization approach

Today, two major trends of sheet metal forming are the improvement of process control and the customization of semi-finished products. Both trends are intertwined, as modified wrought materials introduce new failure modes to the forming process and altered boundary conditions to the determination of optimal process control. The objective of this paper is to provide an effective, robust, and reliable optimization approach for process control of sheet metal forming processes with tailored semi-finished parts. For demonstration purposes, the obtained optimization scheme is applied to sheet metal hydroforming with blanks reinforced by laser-welded stringers prior to forming (stringer sheets). Controllable parameters of this process are the internal pressure and the blank holder force. The predominant failure modes, related to the nature of the semi-finished part, are stringer buckling and necking (bursting) at the stringer ends. The related control problem was formulated as an optimization problem with objective function and restrictions on the optimization parameters (process control) and state variables (displacement). The optimization and state variables were coupled by a finite element simulation of the process. The optimization problem was treated by a penalty approach and solved using the COBYLA algorithm (Constrained Optimization BY Linear Approximations) without any need of further user interference. The obtained optimal control of a time-varying blank holder force significantly extends the capabilities of the targeted forming process, as it allows for a defect-free production of parts with narrower curvatures and higher stringers. All simulation and optimization results are validated experimentally.