Substructuring in the implicit simulation of single point incremental sheet forming

This paper presents a direct substructuring method to reduce the computing time of implicit simulations of single point incremental forming (SPIF). Substructuring is used to divide the finite element (FE) mesh into several non-overlapping parts. Based on the hypothesis that plastic deformation is localized, the substructures are categorized into two groups: the plastic—nonlinear—substructures and the elastic—pseudo-linear—substructures. The plastic substructures assemble a part of the FE mesh that is in contact with the forming tool; they are iteratively updated respecting all nonlinearities. The elastic substructures model the elastic deformation of the rest of the FE mesh. For these substructures, the geometrical and the material behaviour are assumed linear within the increment. The stiffness matrices and the internal force vectors are calculated at the beginning of each increment then they are statically condensed to eliminate the internal degrees of freedom (DOF). In the iteration process the condensed stiffness matrices for the elastic substructures are kept constant. The condensed internal force vectors are updated by the multiplication of the condensed stiffness matrices and the displacement increments. After convergence, any geometrical and material nonlinearity for the elastic substructures are nonlinearly updated. The categorization of substructures in plastic and elastic domains is adapted during the simulation to capture the tool motion. The resulting, plastic and condensed elastic, set of equations is solved on a single processor. In an example with 1600 shell elements, the presented substructuring of the SPIF implicit simulation is 2.4 times faster than the classical implicit simulation.