A continuum sensitivity analysis of large deformations with applications to metal forming process design

A continuum sensitivity analysis is presented for the evaluation of parameter and shape sensitivities of large hyperelastic-viscoplastic deformations of isotropic materials involving contact with friction using a direct differentiation method. The sensitivity formulation is based on the design differentiation of the governing field equations of the direct deformation problem at the continuum level and a weak form for the sensitivity equilibrium equation is developed. This sensitivity weak form is linearly coupled with the appropriate sensitivity contact and sensitivity constitutive problems. To avoid issues related to the non-differentiablity of contact conditions, regularizing assumptions are introduced for the computation of traction sensitivities. The sensitivity weak form is modified for the consistent finite element treatment of near-incompressibility within the context of the assumed strain methods. The sensitivity analysis is derived within a total Lagrangian as well an updated Lagrangian framework. The updated Lagrangian sensitivity analysis is appropriate when remeshing operations are performed in the direct deformation problem to avoid the excessive distortions that result in Lagrangian finite element formulations. A method is also proposed for the transfer of design sensitivities between meshes. The results of the continuum sensitivity analysis are used to compute design gradients of the objective function and constraints in an appropriately selected finite dimensional design space, in order to solve selective design problems in metal forming. This work on the design of single-stage forming processes is expanded to include the preliminary design of multi-stage forming processes.