# Crystal plasticity with Jacobian-Free Newton–Krylov

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DOI: 10.1007/s00466-012-0741-7

- Cite this article as:
- Chockalingam, K., Tonks, M.R., Hales, J.D. et al. Comput Mech (2013) 51: 617. doi:10.1007/s00466-012-0741-7

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## Abstract

The objective of this work is to study potential benefits of solving crystal plasticity finite element method (CPFEM) implicit simulations using the Jacobian-Free Newton–Krylov (JFNK) technique. Implicit implementations of CPFEM are usually solved using Newton’s method. However, the inherent non-linearity in the flow rule model that characterizes the crystal slip system deformation on occasions would require considerable effort to form the exact analytical Jacobian needed by Newton’s method. In this paper we present an alternative using JFNK. As it does not require an exact Jacobian, JFNK can potentially decrease development time. JFNK approximates the effect of the Jacobian through finite differences of the residual vector, allowing modified formulations to be studied with relative ease. We show that the JFNK solution is identical to that obtained using Newton’s method and produces quadratic convergence. We also find that preconditioning the JFNK solution with the elastic tensor provides the best computational efficiency.