Stamping process design using shape optimization techniques based on adjoint methods

Many design processes could be made faster and cheaper by use of shape optimization techniques. However, the computational cost of evaluating the necessary derivatives often limits the application of optimization methods to simple engineering problems. Nevertheless, techniques based on the control theory have been shown to decrease dramatically this cost when compared with Finite Difference methods. In this paper we describe how an optimization algorithm can be used, along with adjoint based methods, in the stamping process design. We first show how the stamping process can be stated in mathematical terms and expressed as a classical optimization problem. In this formulation, the objective and constraint functions are expressed in terms of the strain distribution in the part. This strain distribution is evaluated by a simplified finite element procedure. We then show how the gradients and Hessians can be computed with adjoint based methods. In our approach we use the shape of the entire addendum surface as the design variable. This surface is approximated by the finite element mesh and the discrete design variables are the vertical coordinates of the nodes. Finally, plane strain examples illustrate the model described.