A continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design problems

A continuum sensitivity analysis is presented for large inelastic deformations and metal forming processes. The formulation is based on the di erentiation of the governing eld equations of the direct problem and development of weak forms for the corresponding eld sensitivity equations. Special attention is given to modelling of the sensitivity boundary conditions that result due to frictional contact between the die and the workpiece. The contact problem in the direct deformation analysis is modelled using an augmented Lagrangian formulation. To avoid issues of non-di erentiability of the contact conditions, appropriate regular- izing assumptions are introduced for the calculation of the sensitivity of the contact tractions. The proposed analysis is used for the calculation of sensitivity elds with respect to various process parameters including the die surface. The accuracy and e ectiveness of the proposed method are demonstrated with a number of representative example problems. In the die design applications, a B ezier representation of the die curve is introduced. The control points of the B ezier curve are used as the design parameters. Comparison of the computed sensitivity results with those obtained using the direct analysis for two nearby dies and a nite di erence approximation indicate a very high accuracy of the proposed analysis. The method is applied to the design of extrusion dies that minimize the standard deviation of the material state in the nal product or minimize the required extrusion force for a given reduction ratio. An open-forging die is also designed which for a speci ed stroke and initial workpiece produces a nal product of desired shape.