Implicit integration methods for dislocation dynamics
Abstract
In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events, and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. Here, this paper investigates the viability of high order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge-Kutta time integrators are explored as a way of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Southern Methodist Univ., Dallas, TX (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1266679
- Report Number(s):
- LLNL-JRNL-656343
Journal ID: ISSN 0965-0393
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Modelling and Simulation in Materials Science and Engineering
- Additional Journal Information:
- Journal Volume: 23; Journal Issue: 2; Journal ID: ISSN 0965-0393
- Publisher:
- IOP Publishing
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 36 MATERIALS SCIENCE
Citation Formats
Gardner, D. J., Woodward, C. S., Reynolds, D. R., Hommes, G., Aubry, S., and Arsenlis, A. Implicit integration methods for dislocation dynamics. United States: N. p., 2015.
Web. doi:10.1088/0965-0393/23/2/025006.
Gardner, D. J., Woodward, C. S., Reynolds, D. R., Hommes, G., Aubry, S., & Arsenlis, A. Implicit integration methods for dislocation dynamics. United States. https://doi.org/10.1088/0965-0393/23/2/025006
Gardner, D. J., Woodward, C. S., Reynolds, D. R., Hommes, G., Aubry, S., and Arsenlis, A. Tue .
"Implicit integration methods for dislocation dynamics". United States. https://doi.org/10.1088/0965-0393/23/2/025006. https://www.osti.gov/servlets/purl/1266679.
@article{osti_1266679,
title = {Implicit integration methods for dislocation dynamics},
author = {Gardner, D. J. and Woodward, C. S. and Reynolds, D. R. and Hommes, G. and Aubry, S. and Arsenlis, A.},
abstractNote = {In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events, and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. Here, this paper investigates the viability of high order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge-Kutta time integrators are explored as a way of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.},
doi = {10.1088/0965-0393/23/2/025006},
journal = {Modelling and Simulation in Materials Science and Engineering},
number = 2,
volume = 23,
place = {United States},
year = {Tue Jan 20 00:00:00 EST 2015},
month = {Tue Jan 20 00:00:00 EST 2015}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 14 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Hybrid Krylov Methods for Nonlinear Systems of Equations
journal, May 1990
- Brown, Peter N.; Saad, Youcef
- SIAM Journal on Scientific and Statistical Computing, Vol. 11, Issue 3
CVODE, A Stiff/Nonstiff ODE Solver in C
journal, January 1996
- Cohen, Scott D.; Hindmarsh, Alan C.; Dubois, Paul F.
- Computers in Physics, Vol. 10, Issue 2
Simulation of dislocations on the mesoscopic scale. I. Methods and examples
journal, January 1999
- Schwarz, K. W.
- Journal of Applied Physics, Vol. 85, Issue 1
Efficient time integration in dislocation dynamics
journal, January 2014
- Sills, Ryan B.; Cai, Wei
- Modelling and Simulation in Materials Science and Engineering, Vol. 22, Issue 2
Two classes of multisecant methods for nonlinear acceleration
journal, March 2009
- Fang, Haw-ren; Saad, Yousef
- Numerical Linear Algebra with Applications, Vol. 16, Issue 3
Inexact Newton Methods
journal, April 1982
- Dembo, Ron S.; Eisenstat, Stanley C.; Steihaug, Trond
- SIAM Journal on Numerical Analysis, Vol. 19, Issue 2
A multiscale strength model for extreme loading conditions
journal, April 2011
- Barton, N. R.; Bernier, J. V.; Becker, R.
- Journal of Applied Physics, Vol. 109, Issue 7
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986
- Saad, Youcef; Schultz, Martin H.
- SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
book, January 1998
- Ascher, Uri M.; Petzold, Linda R.
Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications
journal, November 1998
- Verdier, M.; Fivel, M.; Groma, I.
- Modelling and Simulation in Materials Science and Engineering, Vol. 6, Issue 6
An accelerated Picard method for nonlinear systems related to variably saturated flow
journal, March 2012
- Lott, P. A.; Walker, H. F.; Woodward, C. S.
- Advances in Water Resources, Vol. 38
Numerical simulation of squeeze cast magnesium alloy AZ91D
journal, October 2001
- Hu, Henry; Yu, Alfred
- Modelling and Simulation in Materials Science and Engineering, Vol. 10, Issue 1
Choosing the Forcing Terms in an Inexact Newton Method
journal, January 1996
- Eisenstat, Stanley C.; Walker, Homer F.
- SIAM Journal on Scientific Computing, Vol. 17, Issue 1
Enabling strain hardening simulations with dislocation dynamics
journal, July 2007
- Arsenlis, A.; Cai, W.; Tang, M.
- Modelling and Simulation in Materials Science and Engineering, Vol. 15, Issue 6
A fast algorithm for particle simulations
journal, December 1987
- Greengard, L.; Rokhlin, V.
- Journal of Computational Physics, Vol. 73, Issue 2
SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers
journal, September 2005
- Hindmarsh, Alan C.; Brown, Peter N.; Grant, Keith E.
- ACM Transactions on Mathematical Software, Vol. 31, Issue 3
Iterative Procedures for Nonlinear Integral Equations
journal, October 1965
- Anderson, Donald G.
- Journal of the ACM, Vol. 12, Issue 4
Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation
journal, January 2000
- Ghoniem, N. M.; Tong, S. -H.; Sun, L. Z.
- Physical Review B, Vol. 61, Issue 2
Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension
journal, May 1998
- Carlson, Neil N.; Miller, Keith
- SIAM Journal on Scientific Computing, Vol. 19, Issue 3
Additive Runge–Kutta schemes for convection–diffusion–reaction equations
journal, January 2003
- Kennedy, Christopher A.; Carpenter, Mark H.
- Applied Numerical Mathematics, Vol. 44, Issue 1-2
Anderson Acceleration for Fixed-Point Iterations
journal, January 2011
- Walker, Homer F.; Ni, Peng
- SIAM Journal on Numerical Analysis, Vol. 49, Issue 4
A Fast Algorithm for Particle Simulations
journal, August 1997
- Greengard, L.; Rokhlin, V.
- Journal of Computational Physics, Vol. 135, Issue 2
Works referencing / citing this record:
Fundamentals of Dislocation Dynamics Simulations
book, January 2016
- Sills, Ryan B.; Kuykendall, William P.; Aghaei, Amin
- Multiscale Materials Modeling for Nanomechanics
Advanced time integration algorithms for dislocation dynamics simulations of work hardening
journal, April 2016
- Sills, Ryan B.; Aghaei, Amin; Cai, Wei
- Modelling and Simulation in Materials Science and Engineering, Vol. 24, Issue 4