Conventional crystal plasticity (CP) solvers are based on a Newton-Raphson (NR) approach which
use an initial guess for the free variables (often stress) to be solved. These solvers are limited by a
finite interval of convergence and often fail when the free variable falls outside this interval.
Solution failure results in the reduction of the time increment to be solved, thus convergence of
the CP solver is a bottleneck which determines the computational cost of the simulation. The
numerical stability of the slip law in its inverted form offers a solver that isn’t vulnerable to poor
pre-conditioning (initial guess) and can be used to progress to a solution from a stable starting
point (i.e., from zero slip rate γ˙
k
p = 0 s
− 1
). In this paper, a novel formulation that enables the
application of the slip law in its inverted form is introduced; this treats all slip systems as independent by approximating the Jacobian as a diagonal matrix, thus overcomes ill-defined and
singular Jacobians associated with previous approaches. This scheme was demonstrated to offer
superior robustness and convergence rate for a case with a single slip system, however the
convergence rate for extreme cases with several active slip systems was relatively poor. Here, we
introduce a novel ‘hybrid scheme’ that first uses the reverse scheme for the first stage of the
solution, and then transitions to the forward scheme to complete the solution at a higher
convergence rate. Several examples are given for pointwise calculations, followed by CPFEM
simulations for FCC copper and HCP Zircaloy-4, which demonstrated solver performance in
practise. The performance of simulations using the hybrid scheme was shown to require six to
nine times fewer increments compared to the conventional forward scheme solver based on a free
variable of stress and initial guess based on a fully elastic increment.