Modular implementation framework of partitioned path-following strategies: Formulation, algorithms and application to the finite element software Cast3M

Abstract Damage, cracking, and strain localization mechanisms often lead to unstable structural responses characterized by snap-backs (i.e., force and displacement decrease simultaneously). Standard nonlinear Newton-based solution algorithms with displacement/force control cannot capture the equilibrium curve in its entirety. This can be overcome using path-following formulations. A general (i.e., valid for any finite element code) implementation framework can be designed and applied to the Cast3M software by collecting the essential concepts and the formalism of partitioned path-following arc-length algorithms. Thanks to these developments, Cast3M is now capable of processing path-following equations without any major modifications. Three path-following constraints were selected to demonstrate the applicability of this framework: a first one on the combination of the displacement increment at a given set of nodes, a second one on the maximum strain increment over the computational domain, and a third one on the maximum elastic predictor of the damage/plastic criterion function over the computational domain. Two- and three-dimensional strain localization simulations show that the proposed framework behaves in a stable and convergent manner, even when multiple severe snap-back instabilities are present. Users of Cast3M shall find the proposed study helpful in that it allows them to focus on developing new path-following equations for the software. Cast3M is developed by the French Alternative Energies and Atomic Energy Commission (CEA) and freely available for research purposes. The developments discussed in this paper have been made available to the user/developer community along with Cast3M 2021 (release date: June 2021).