A sub-stepping scheme for multi-scale analysis of solids

This paper proposes a sub-stepping procedure aimed at saving overall computing time in the finite element solution of non-linear solid mechanics problems where homogenization-based multi-scale constitutive models are used to describe the material response. The procedure obtains better initial guesses for the Newton–Raphson scheme adopted in the solution of the incremental equilibrium problem of the representative volume element (RVE) associated with each Gauss quadrature point of the discretised macroscopic continuum. The resulting gain in robustness of the RVE Newton–Raphson iteration allows larger time/load steps to be prescribed at the macroscopic scale, leading to substantial savings in computing time. The proposed sub-stepping preserves the quadratic rate of asymptotic convergence of the macroscopic scale Newton–Raphson scheme, without compromising the accuracy of the solution. Numerical examples demonstrate that the achieved improvement in efficiency is particularly noticeable under the assumption of finite stains, where overall speed-up factors of up to four are observed.