Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

A design sensitivity analysis scheme is proposed for an elasto-plastic finite element analysis with explicit time integration using a direct differentiation method (DDM). The direct differentiation scheme proposed deals with large deformation, an elasto-plastic constitutive relation considering the planar anisotropy, shell elements with reduced integration and complicated contact between the sheet and the dies. The design sensitivities with respect to the process parameter are calculated with direct analytical differentiation of the governing equation. The present DDM result is compared with the result obtained from a finite difference method (FDM) in sheet metal forming processes such as a cylindrical cup drawing process and a U draw-bending process for the verification of its reliability, accuracy and versatility. DDM results show good agreement with FDM results. Results fully demonstrate that sensitivity calculation with FDM is sometimes very dangerous since it could lead to a fatal pitfall and selection of the sensitivity scheme should be carefully done for specific problems. The analysis shows that the sensitivity of state variables provides useful information not only for the strain control but also for the inspection of defect initiation such as fracture or wrinkle. The result demonstrates that the sensitivity calculation scheme proposed is applicable to analysis and design of complicated sheet metal forming.