A continuum sensitivity method for the design of multi-stage metal forming processes

A novel, efficient and mathematically rigorous continuum based sensitivity method is introduced that can be used to accurately evaluate the gradients of the objective function and constraints in the optimization-based design of multi-stage deformation processes. Weak sensitivity equilibrium equations are derived for the large deformation of the workpiece in each forming operation. This sensitivity kinematic problem is linearly coupled with the corresponding continuum sensitivity constitutive, contact and thermal sub-problems for the particular process. Thus a linear sensitivity problem with appropriate driving forces is identified and the analysis is carried out in an infinite dimensional framework. The multi-stage continuum sensitivity analysis takes a form similar to the updated Lagrangian sensitivity framework developed earlier for the design of single-stage deformation processes. It allows us to treat in a unified manner shape and parameter sensitivity analyses that are both present in a typical design problem of multi-stage deformation processes. The effectiveness of the proposed methodology is demonstrated with the solution of three practical problems in the design of two-stage metal forming processes.