Shape optimization of axisymmetric preform tools in forging using a direct differentiation method

A method for computing shape sensitivity in the frame of non-linear and non-steady-state forging is presented. Derivatives of tool geometry, velocity and state variables with respect to the shape parameters are calculated by a direct di3erentiation of discrete equations. Because of the important part played by the accuracy of =nite element calculations, an e>cient transfer method is used between meshes during remeshings and the contact algorithms are carefully di3erentiated. The resulting inverse design procedure is successfully applied to two industrial examples of forging of automobile parts, with fold-over and piping defects occurring during the intermediate designs. It makes it possible to suggest reasonable preform shapes, with or without any available knowledge of the forging process