Shape optimization and preform design in metal forming processes Original Research Article

A continuum sensitivity analysis is presented for the computation of the shape sensitivity of finite hyperelastic–viscoplastic deformations involving contact with friction using a direct differentiation method. Weak shape sensitivity equations are developed that are consistent with the kinematic analysis, constitutive sub-problem as well as the analysis of the contact/friction sub-problem used in the solution of the direct deformation problem. The shape sensitivities are defined in a rigorous sense and the linear sensitivity analysis is performed in an infinite-dimensional continuum framework. The direct deformation and the sensitivity deformation problems are implemented using the finite element method. The shape sensitivity analysis is validated by a comparison of the results with those obtained from the solution of the perturbed direct deformation problem (i.e. using finite differences). Finite-dimensional gradients of objective functions are then computed using the results of the shape sensitivity analysis for the purpose of preform design and shape optimization in metal forming. The effectiveness of the proposed methodology is demonstrated by solving practical shape optimal design problems in forging processing.