Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media

Chen, C.; Aubry, S.; Oppelstrup, T.; Arsenlis, A.; Darve, E.

doi:10.1088/1361-651X/aab7bb

Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media

Journal Article · Mon Mar 19 00:00:00 EDT 2018 · Modelling and Simulation in Materials Science and Engineering

DOI:https://doi.org/10.1088/1361-651X/aab7bb· OSTI ID:1458681

^[1]; Aubry, S. ^[2]; Oppelstrup, T. ^[2]; Arsenlis, A. ^[2]; Darve, E. ^[1]

Stanford Univ., CA (United States)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). In this study, we implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Lastly, various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1458681

Report Number(s):: LLNL-JRNL--742868; 897727

Journal Information:: Modelling and Simulation in Materials Science and Engineering, Journal Name: Modelling and Simulation in Materials Science and Engineering Journal Issue: 4 Vol. 26; ISSN 0965-0393

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United States

Language:: English

References (52)

Fast solution method for three-dimensional Stokesian many-particle problems Fu, Yuhong; Rodin, Gregory J. Communications in Numerical Methods in Engineering, Vol. 16, Issue 2 https://doi.org/10.1002/(sici)1099-0887(200002)16:2<145::aid-cnm323>3.0.co;2-e	journal	February 2000
Green's functions for anisotropic elasticity Mura, T.; Kinoshita, N. Physica Status Solidi (b), Vol. 47, Issue 2 https://doi.org/10.1002/pssb.2220470226	journal	October 1971
Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions Rokhlin, V. Applied and Computational Harmonic Analysis, Vol. 1, Issue 1 https://doi.org/10.1006/acha.1993.1006	journal	December 1993
A Fast Algorithm for Particle Simulations Greengard, L.; Rokhlin, V. Journal of Computational Physics, Vol. 135, Issue 2 https://doi.org/10.1006/jcph.1997.5706	journal	August 1997
Yet Another Fast Multipole Method without Multipoles—Pseudoparticle Multipole Method Makino, Junichiro Journal of Computational Physics, Vol. 151, Issue 2 https://doi.org/10.1006/jcph.1999.6226	journal	May 1999
The Fast Multipole Method: Numerical Implementation Darve, Eric Journal of Computational Physics, Vol. 160, Issue 1 https://doi.org/10.1006/jcph.2000.6451	journal	May 2000
A fast algorithm for particle simulations Greengard, L.; Rokhlin, V. Journal of Computational Physics, Vol. 73, Issue 2 https://doi.org/10.1016/0021-9991(87)90140-9	journal	December 1987
Rapid solution of integral equations of scattering theory in two dimensions Rokhlin, V. Journal of Computational Physics, Vol. 84, Issue 1 https://doi.org/10.1016/0021-9991(89)90191-5	journal	September 1989
Rapid solution of integral equations of scattering theory in two dimensions Rokhlin, V. Journal of Computational Physics, Vol. 86, Issue 2 https://doi.org/10.1016/0021-9991(90)90107-C	journal	February 1990
Rapid solution of integral equations of scattering theory in two dimensions Rokhlin, V. Journal of Computational Physics, Vol. 86, Issue 2 https://doi.org/10.1016/0021-9991(90)90107-c	journal	February 1990
Dislocation distributions in two dimensions Gulluoglu, A. N.; Srolovitz, D. J.; LeSar, R. Scripta Metallurgica, Vol. 23, Issue 8 https://doi.org/10.1016/0036-9748(89)90057-4	journal	August 1989
Fast multiplication of a recursive block Toeplitz matrix by a vector and its application Lee, David Journal of Complexity, Vol. 2, Issue 4 https://doi.org/10.1016/0885-064X(86)90007-5	journal	December 1986
Fast multiplication of a recursive block Toeplitz matrix by a vector and its application Lee, David Journal of Complexity, Vol. 2, Issue 4 https://doi.org/10.1016/0885-064x(86)90007-5	journal	December 1986
On plastic deformation and the dynamics of 3D dislocations Zbib, Hussein M.; Rhee, Moono; Hirth, John P. International Journal of Mechanical Sciences, Vol. 40, Issue 2-3 https://doi.org/10.1016/S0020-7403(97)00043-X	journal	February 1998
Mesoscopic simulations of dislocations and plasticity Devincre, B.; Kubin, L. P. Materials Science and Engineering: A, Vol. 234-236 https://doi.org/10.1016/S0921-5093(97)00146-9	journal	August 1997
A dislocation dynamics study of the transition from homogeneous to heterogeneous deformation in irradiated body-centered cubic iron Arsenlis, A.; Rhee, M.; Hommes, G. Acta Materialia, Vol. 60, Issue 9 https://doi.org/10.1016/j.actamat.2012.03.041	journal	May 2012
A kernel-independent adaptive fast multipole algorithm in two and three dimensions Ying, Lexing; Biros, George; Zorin, Denis Journal of Computational Physics, Vol. 196, Issue 2 https://doi.org/10.1016/j.jcp.2003.11.021	journal	May 2004
A parallel algorithm for 3D dislocation dynamics Wang, Zhiqiang; Ghoniem, Nasr; Swaminarayan, Sriram Journal of Computational Physics, Vol. 219, Issue 2 https://doi.org/10.1016/j.jcp.2006.04.005	journal	December 2006
The black-box fast multipole method Fong, William; Darve, Eric Journal of Computational Physics, Vol. 228, Issue 23 https://doi.org/10.1016/j.jcp.2009.08.031	journal	December 2009
A non-singular continuum theory of dislocations Cai, W.; Arsenlis, A.; Weinberger, C. Journal of the Mechanics and Physics of Solids, Vol. 54, Issue 3 https://doi.org/10.1016/j.jmps.2005.09.005	journal	March 2006
Comparing the strength of f.c.c. and b.c.c. sub-micrometer pillars: Compression experiments and dislocation dynamics simulations Greer, Julia R.; Weinberger, Christopher R.; Cai, Wei Materials Science and Engineering: A, Vol. 493, Issue 1-2 https://doi.org/10.1016/j.msea.2007.08.093	journal	October 2008
On plastic deformation and the dynamics of 3D dislocations Zbib, Hussein M.; Rhee, Moono; Hirth, John P. International Journal of Mechanical Sciences, Vol. 40, Issue 2-3 https://doi.org/10.1016/s0020-7403(97)00043-x	journal	February 1998
Mesoscopic simulations of dislocations and plasticity Devincre, B.; Kubin, L. P. Materials Science and Engineering: A, Vol. 234-236 https://doi.org/10.1016/s0921-5093(97)00146-9	journal	August 1997
A new version of the Fast Multipole Method for the Laplace equation in three dimensions Greengard, Leslie; Rokhlin, Vladimir Acta Numerica, Vol. 6 https://doi.org/10.1017/S0962492900002725	journal	January 1997
A multiscale strength model for extreme loading conditions Barton, N. R.; Bernier, J. V.; Becker, R. Journal of Applied Physics, Vol. 109, Issue 7 https://doi.org/10.1063/1.3553718	journal	April 2011
Simulation of dislocations on the mesoscopic scale. I. Methods and examples Schwarz, K. W. Journal of Applied Physics, Vol. 85, Issue 1 https://doi.org/10.1063/1.369429	journal	January 1999
O( N ) algorithm for dislocation dynamics Wang, H. Y.; Lesar, R. Philosophical Magazine A, Vol. 71, Issue 1 https://doi.org/10.1080/01418619508242962	journal	January 1995
Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics Weygand, D.; Friedman, L. H.; Giessen, E. Van der Modelling and Simulation in Materials Science and Engineering, Vol. 10, Issue 4 https://doi.org/10.1088/0965-0393/10/4/306	journal	June 2002
Enabling strain hardening simulations with dislocation dynamics Arsenlis, A.; Cai, W.; Tang, M. Modelling and Simulation in Materials Science and Engineering, Vol. 15, Issue 6 https://doi.org/10.1088/0965-0393/15/6/001	journal	July 2007
A new version fast multipole method for evaluating the stress field of dislocation ensembles Zhao, Degang; Huang, Jingfang; Xiang, Yang Modelling and Simulation in Materials Science and Engineering, Vol. 18, Issue 4 https://doi.org/10.1088/0965-0393/18/4/045006	journal	March 2010
Use of spherical harmonics for dislocation dynamics in anisotropic elastic media Aubry, S.; Arsenlis, A. Modelling and Simulation in Materials Science and Engineering, Vol. 21, Issue 6 https://doi.org/10.1088/0965-0393/21/6/065013	journal	August 2013
Multipole expansion of dislocation interactions: Application to discrete dislocations LeSar, R.; Rickman, J. M. Physical Review B, Vol. 65, Issue 14 https://doi.org/10.1103/PhysRevB.65.144110	journal	April 2002
Multipole representation of the elastic field of dislocation ensembles Wang, Zhiqiang; Ghoniem, Nasr; LeSar, Richard Physical Review B, Vol. 69, Issue 17 https://doi.org/10.1103/PhysRevB.69.174102	journal	May 2004
Multipole expansion of dislocation interactions: Application to discrete dislocations LeSar, R.; Rickman, J. M. Physical Review B, Vol. 65, Issue 14 https://doi.org/10.1103/physrevb.65.144110	journal	April 2002
Multipole representation of the elastic field of dislocation ensembles Wang, Zhiqiang; Ghoniem, Nasr; LeSar, Richard Physical Review B, Vol. 69, Issue 17 https://doi.org/10.1103/physrevb.69.174102	journal	May 2004
On the Compression of Low Rank Matrices Cheng, H.; Gimbutas, Z.; Martinsson, P. G. SIAM Journal on Scientific Computing, Vol. 26, Issue 4 https://doi.org/10.1137/030602678	journal	January 2005
An Accelerated Kernel-Independent Fast Multipole Method in One Dimension Martinsson, P. G.; Rokhlin, V. SIAM Journal on Scientific Computing, Vol. 29, Issue 3 https://doi.org/10.1137/060662253	journal	January 2007
Fast Directional Multilevel Algorithms for Oscillatory Kernels Engquist, Björn; Ying, Lexing SIAM Journal on Scientific Computing, Vol. 29, Issue 4 https://doi.org/10.1137/07068583X	journal	January 2007
Fast Directional Multilevel Algorithms for Oscillatory Kernels Engquist, Björn; Ying, Lexing SIAM Journal on Scientific Computing, Vol. 29, Issue 4 https://doi.org/10.1137/07068583x	journal	January 2007
Fast Algorithms for Polynomial Interpolation, Integration, and Differentiation Dutt, A.; Gu, M.; Rokhlin, V. SIAM Journal on Numerical Analysis, Vol. 33, Issue 5 https://doi.org/10.1137/0733082	journal	October 1996
An Implementation of the Fast Multipole Method without Multipoles Anderson, Christopher R. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 4 https://doi.org/10.1137/0913055	journal	July 1992
Cauchy Fast Multipole Method for General Analytic Kernels Létourneau, Pierre-David; Cecka, Cris; Darve, Eric SIAM Journal on Scientific Computing, Vol. 36, Issue 2 https://doi.org/10.1137/120891617	journal	January 2014
An Improved Fast Multipole Algorithm for Potential Fields on the Line Yarvin, Norman; Rokhlin, Vladimir SIAM Journal on Numerical Analysis, Vol. 36, Issue 2 https://doi.org/10.1137/S0036142997329232	journal	January 1999
The Fast Multipole Method I: Error Analysis and Asymptotic Complexity Darve, Eric SIAM Journal on Numerical Analysis, Vol. 38, Issue 1 https://doi.org/10.1137/S0036142999330379	journal	January 2000
A Generalized Fast Multipole Method for Nonoscillatory Kernels Gimbutas, Zydrunas; Rokhlin, Vladimir SIAM Journal on Scientific Computing, Vol. 24, Issue 3 https://doi.org/10.1137/S1064827500381148	journal	January 2003
Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators Yarvin, N.; Rokhlin, V. SIAM Journal on Scientific Computing, Vol. 20, Issue 2 https://doi.org/10.1137/S1064827596310779	journal	January 1998
An Improved Fast Multipole Algorithm for Potential Fields on the Line Yarvin, Norman; Rokhlin, Vladimir SIAM Journal on Numerical Analysis, Vol. 36, Issue 2 https://doi.org/10.1137/s0036142997329232	journal	January 1999
The Fast Multipole Method I: Error Analysis and Asymptotic Complexity Darve, Eric SIAM Journal on Numerical Analysis, Vol. 38, Issue 1 https://doi.org/10.1137/s0036142999330379	journal	January 2000
A Generalized Fast Multipole Method for Nonoscillatory Kernels Gimbutas, Zydrunas; Rokhlin, Vladimir SIAM Journal on Scientific Computing, Vol. 24, Issue 3 https://doi.org/10.1137/s1064827500381148	journal	January 2003
Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators Yarvin, N.; Rokhlin, V. SIAM Journal on Scientific Computing, Vol. 20, Issue 2 https://doi.org/10.1137/s1064827596310779	journal	January 1998
Dislocation Microstructures and Plastic Flow: A 3D Simulation Kubin, Ladislas P.; Canova, G.; Condat, M. Solid State Phenomena, Vol. 23-24 https://doi.org/10.4028/www.scientific.net/SSP.23-24.455	journal	January 1992
Dislocation Microstructures and Plastic Flow: A 3D Simulation Kubin, Ladislas P.; Canova, G.; Condat, M. Solid State Phenomena, Vol. 23-24 https://doi.org/10.4028/www.scientific.net/ssp.23-24.455	journal	January 1992

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Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media

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