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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Roters F, Eisenlohr P, Bieler TR, Raabe D (2010) Crystal plasticity finite element methods. Wiley-VCH, Weinheim
Roters F, Eisenlohr P, Hantcherli L, Tjahjanto DD, Bieler TR, Raabe D (2010) Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications. Acta Mater 58(4): 1152–1211
Anand L, Kothari M (1996) A computational procedure for rate-independent crystal plasticity. J Mech Phys Solids 44(4): 525–558
Asaro RJ, Needleman A (1985) Overview no. 42 texture development and strain hardening in rate dependent polycrystals. Acta Metall 33(6): 923–953
Huang Y (1991) A user-material subroutine incorporating single crystal plasticity in the abaqus finite element program. Harvard University Report, Technical report
Kothari M, Anand L (1998) Elasto-viscoplastic constitutive equations for polycrystalline metals: application to tantalum. J Mech Phys Solids 46(1): 51–83
Staroselsky A, Anand L (2003) A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy az31b. Int J Plast 19(10): 1843–1864
Hughes TJR, Liu WK (1978) Implicit–explicit finite elements in transient analysis: stability theory. J Appl Mech 45(2): 371–374
Harewood FJ, McHugh PE (2007) Comparison of the implicit and explicit finite element methods using crystal plasticity. Comput Mater Sci 39(2): 481–494
Dokainish MA, Subbaraj K (1989) A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods. Comput Struct 32(6): 1371–1386
Subbaraj K, Dokainish MA (1989) A survey of direct time-integration methods in computational structural dynamics—II. Implicit methods. Comput Struct 32(6): 1387–1401
Cuitino AM, Ortiz M (1992) Computational modelling of single crystals. Model Simul Mater Sci Eng 1(3): 225–263
Segurado J, Lebensohn RA, LLorca J, Tom CN (2012) Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements. Int J Plast 28(1): 124–140
Steinmann P, Stein E (1996) On the numerical treatment and analysis of finite deformation ductile single crystal plasticity. Comput Methods Appl Mech Eng 129(3): 235–254
Kalidindi SR, Bronkhorst CA, Anand L (1992) Crystallographic texture evolution in bulk deformation processing of fcc metals. J Mech Phys Solids 40(3): 537–569
Knoll DA, Keyes DE (2004) Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J Comput Phys 193(2): 357–397
Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving linear systems. SIAM J Sci Stat Comput 7: 856
Hales JD, Novascone SR, Williamson RL, Gaston DR, Tonks MR (2012) Solving nonlinear solid mechanics problems with the Jacobian-free Newton–Krylov method. Comput Model Eng Sci 84(2): 123–154
Gaston D, Newman C, Hansen G, Lebrun-Grandi D (2009) Moose: a parallel computational framework for coupled systems of nonlinear equations. Nucl Eng Des 239(10): 1768–1778
Busso EP, McClintock FA (1996) A dislocation mechanics-based crystallographic model of a b2-type intermetallic alloy. Int J Plast 12(1): 1–28
Meissonnier FT, Busso EP, O’Dowd NP (2001) Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains. Int J Plast 17(4): 601–640
Tonks MR, Bingert JF, Bronkhorst CA, Harstad EN, Tortorelli DA (2009) Two stochastic mean-field polycrystal plasticity methods. J Mech Phys Solids 57(8): 1230–1253
Bronkhorst CA, Hansen BL, Cerreta EK, Bingert JF (2007) Modeling the microstructural evolution of metallic polycrystalline materials under localization conditions. J Mech Phys Solids 55(11): 2351–2383
Acharya A, Beaudoin AJ (2000) Grain-size effect in viscoplastic polycrystals at moderate strains. J Mech Phys Solids 48(10): 2213–2230
Kocks UF (1976) Laws for work-hardening and low-temperature creep. J Eng Mater Technol 98(1): 76–85
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chockalingam, K., Tonks, M.R., Hales, J.D. et al. Crystal plasticity with Jacobian-Free Newton–Krylov. Comput Mech 51, 617–627 (2013). https://doi.org/10.1007/s00466-012-0741-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-012-0741-7