A robust and efficient hybrid solver for crystal plasticity

• Published in: International journal of plasticity 2023-11, Vol.170, p.103773, Article 103773
• Main Authors: Hardie, Chris, Long, Daniel J., Demir, Eralp, Tarleton, Edmund, Dunne, Fionn P.E.
• Format: Article
• Language: English
• Subjects: B. Crystal plasticity; B. Elastic-viscoplastic material; B. Rate-dependent material; C. Finite element; C. Numerical algorithms
• ISSN: 0749-6419; 1879-2154
• DOI: 10.1016/j.ijplas.2023.103773

•A new solver is introduced, which uses the slip law in its inverted form.•The hybrid scheme was found to offer robustness and efficiency.•CPFEM simulations for FCC copper and HCP Zircaloy-4 were investigated.•A simulation of a challenging problem was 6–9 faster than a conventional solver.•The new solver is easy to implement into existing crystal plasticity code. 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˙pk=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.