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

A novel, e/cient 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 identi7ed and the analysis is carried out in an in7nite 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 uni7ed manner shape and parameter sensitivity analyses that are both present in a typical design problem of multi-stage deformation processes. The e8ectiveness of the proposed methodology is demonstrated with the solution of three practical problems in the design of two-stage metal forming processes. ? 2003 Elsevier Science Ltd. All rights reserved.