Optimal Control of Hydroforming Processes Based on Reduced Order Models

Daniela Koller* and Stefan Ulbrich

The manufacturing of load-carrying sheet metal structures with complex curvatures and stringers from a single workpiece (integral design) is a challenging task. A new approach is to create flat sheet metal parts with stringers and to use them as wrought material for a subsequent forming step, which is in our case the sheet metal hydroforming. With this approach it is possible to overcome the various drawbacks of the state of the art manufacturing procedures. Sheet metal hydroforming is a deep drawing process where the punch is replaced by a pressurized fluid medium, which performs the forming operation. The hydroforming process is affected by the pressure of the fluid medium and the blank holder force. We will present an optimal control problem for the hydroforming process where process limitations resulting from the use of sheet metal parts with stringers are incorporated as constraints. Numerical results for an engineering application, a rectangular cup, will be shown. \newline Mathematically, the hydroforming process leads to an evolution quasi-variational inequality involving contact, friction and plasticity. The finite element simulation of the hydroforming process is computationally expensive, hence we adopt model reduction techniques. We use Proper Orthogonal Decomposition (POD) and a reduced basis method for the constrained states in combination with a semismooth Newton method to obtain a low-order model of the hydroforming process. Numerical results for an optimal control problem based on the developed reduced order models will be presented.

Mathematics Subject Classification: 90C90 49M15 90C33 74S05

Keywords: optimal control; reduced order models; variational inequality

Minisymposion: Optimization of Mechanical Systems