An adaptive sub-incremental strategy for the solution of homogenization-based multi-scale problems

In this contribution, several schemes for the solution of homogenization-based multi-scale constitutive problems undergoing finite strains with inelastic material behavior are investigated. These schemes are aimed at improving the robustness and efficiency of the Newton–Raphson method in the multilevel finite element (ML-FEM) framework. An adaptive sub-incremental strategy is proposed for the discrete representative volume element (RVE) boundary value problem. The procedure is able to ensure the convergence of the solution algorithm, in the presence of several sources of non-linearity, and obtains improved initial guesses for the Newton–Raphson scheme in the ML-FEM framework. The enlargement of the convergence bowl of the Newton–Raphson procedure at the micro-scale allows larger macroscopic deformation gradients to be prescribed and significantly reduces the overall computational cost of ML-FEM analyses. The proposed strategy preserves the quadratic rates of asymptotic convergence that characterize the Newton–Raphson scheme at the macroscopic level. Numerical examples of both micro-scale and two-scale finite element simulations are presented to demonstrate the improved robustness and efficiency of the solution procedures proposed.