Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T06:08:35.213Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium

Published online by Cambridge University Press:  20 August 2015

Degang Zhao ,
Jingfang Huang  and
Yang Xiang
Degang Zhao*
Affiliation:
Nano Science and Technology Program, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
Jingfang Huang*
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
Yang Xiang*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve the boundary value problem for the stress field associated with dislocations in a finite medium. Numerical examples are presented to examine the influence from material boundaries on dislocations.

Keywords

74S1565R2074A1074B05Fast multipole methodboundary integral equation methoddislocation dynamicsstress
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 226 - 246
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Hirth, J. P. and Lothe, J., Theory of Dislocations, 2nd ed., John Wiley, New York, 1982.Google Scholar
[2]Gullouglu, A. N., Srolovitz, D. J., LeSar, R. and Lomdahl, P. S., Dislocation distributions in two dimensions, Scripta Metall., 23 (1989), 1347–1352.Google Scholar
[3]Kubin, L. P., Canova, G., Condat, M., Devincre, B., Pontikis, V. and Brechet, Y., Dislocation microstructures and plastic flow: a 3D simulation, Solid State Phenom., 23/24 (1992), 455–472.Google Scholar
[4]Wang, H. Y. and LeSar, R., O(N) algorithm for dislocation dynamics, Phil. Mag. A, 71 (1995), 149–164.Google Scholar
[5]Fivel, M. C., Gosling, T. J. and Canova, G. R., Implementing image stresses in a 3D dislocation simulations, Model. Simul. Mater. Sci. Eng., 4 (1996), 581–596.CrossRefGoogle Scholar
[6]Zbib, H. M., Rhee, M. and Hirth, J. P., On plastic deformation and the dynamics of 3D dislocations, Int. J. Mech. Sci., 40 (1998), 113–127.Google Scholar
[7]Verdier, M., Fivel, M. and Groma, I., Mesoscopic scale simulation of dislocation dynamics in FCC metals: principles and applications, Model. Simul. Mater. Sci. Eng., 6 (1998), 755–770.Google Scholar
[8]Schwarz, K. W., Simulation of dislocations on the mesoscopic scale I: methods and examples, J. Appl. Phys., 85 (1999), 108–119.CrossRefGoogle Scholar
[9]Ghoniem, N. M., Tong, S. H. and Sun, L. Z., Parametric dislocation dynamics: a thermodynamics-based approach to investgations of mesoscopic plastic deformation, Phys. Rev. B, 61 (2000), 913–927.Google Scholar
[10]Weygand, D., Friedman, L. H., Van, E. der Giessen and Needleman, A., Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics, Model. Simul. Mater. Sci. Eng., 10 (2002), 437–468.Google Scholar
[11]LeSar, R. and Rickman, J. M., Multipole expansion of dislocation interactions: application to discrete dislocations, Phys. Rev. B, 65 (2002), 144110.Google Scholar
[12]Martinez, R. and Ghoniem, N. M., The influence of crystal surfaces on dislocation interactions in mesoscopic plasticity: a combined dislocation dynamics-finite element approach, Comput. Model. Eng. Sci., 3 (2002), 229–245.Google Scholar
[13]Khraishi, T. A. and Zbib, H. M., Free-surface effects in 3D dislocation dynamics: formulation and modelling, J. Eng. Mat. Tech., 124 (2002), 342–351.CrossRefGoogle Scholar
[14]Groh, S., Devincre, B., Kubin, L. P., Roos, A., Feyel, F. and Chaboche, J. L., Dislocation and elastic anisotropy in heteroepitaxial metallic thin films, Phil. Mag. Lett., 83 (2003), 303–313.Google Scholar
[15]Xiang, Y., Cheng, L. T., Srolovitz, D. J. and , W. E, A level set method for dislocation dynamics, Acta Mater., 51 (2003), 5499–5518.Google Scholar
[16]Yan, L., Khraishi, T. A., Shen, Y. L. and Horstemeyer, M. F., A distributed dislocation method for treating free-surface image stresses in three dimensional dislocation dynamics simulations, Model. Simul. Mater. Sci. Eng., 12 (2004), S289–S301.CrossRefGoogle Scholar
[17]Wang, Z., Ghoniem, N. M. and LeSar, R., Multipole representation of the elastic field of dislocation ensembles, Phys. Rev. B, 69 (2004), 174102.CrossRefGoogle Scholar
[18]Liu, X. H. and Schwarz, K. W., Modelling of dislocation intersecting a free surface, Model. Simul. Mater. Sci. Eng., 13 (2005), 1233–1247.CrossRefGoogle Scholar
[19]Quek, S. S., Xiang, Y., Zhang, Y. W., Srolovitz, D. J. and Lu, C., Level set simulation of dislocation dynamics in thin films, Acta Mater., 54 (2006), 2371–2381.Google Scholar
[20]Xiang, Y., Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383–424.Google Scholar
[21]Wang, Z., Ghoniem, N. M., Swaminarayan, S. and LeSar, R., A parallel algorithm for 3D dislocation dynamics, J. Comput. Phys., 219 (2006), 608–621.CrossRefGoogle Scholar
[22]Tang, M. J., Cai, W., Xu, G. S. and Bulatov, V. V., A hybrid method for computing forces on curved dislocations intersecting free surfaces in three-dimensional dislocation dynamics, Model. Simul. Mater. Sci. Eng., 14 (2006), 1139–1151.Google Scholar
[23]Arsenlis, A., Cai, W., Tang, M., Rhee, M., Oppelstrup, T., Hommes, G., Pierce, T. G. and Bulatov, V. V., Enabling strain hardening simulations with dislocation dynamics, Model. Simul. Mater. Sci. Eng., 15 (2007), 553–595.Google Scholar
[24] J. El-Awady, A., Biner, S. B. and Ghoniem, N. M., A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes, J. Mech. Phys. Solids, 56 (2008), 2019–2035.Google Scholar
[25]Quek, S. S., Zhang, Y. W., Xiang, Y. and Srolovitz, D. J., Dislocation cross-slip in heteroepitaxial multilayer films, Acta Mater., 58 (2010), 226–234.Google Scholar
[26]Zhao, D. G., Huang, J. F. and Xiang, Y., A new version fast multipole method for evaluating the stress field of dislocation ensembles, Model. Simul. Mater. Sci. Eng., 18 (2010), 045006.Google Scholar
[27]Greengard, L. and Rokhlin, V., A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numer., 6 (1997), 229–269.Google Scholar
[28]Giessen, E. van der and Needleman, A., Discrete dislocation plasticity: a simple planar model, Model. Simul. Mater. Sci. Eng., 3 (1995), 689–735.Google Scholar
[29]Brebbia, C. A., The Boundary Element Method for Engineers, Pentech Press, London, 1978.Google Scholar
[30]Fu, Y., Klimkowski, K. J., Rodin, G. J., Berger, E., Browne, J. C., Singer, J. K., Geijn, R. A. van de and Vemaganti, K. S., A fast solution method for three-dimensional many-particle problems of linear elasticity, Int. J. Numer. Meth. Eng., 42 (1998), 1215–1229.Google Scholar
[31]Fu, Y. and Rodin, G. J., Fast solution method for three-dimensional Stokesian many-particle problems, Commun. Numer. Meth. Eng., 16 (2000), 145–149.Google Scholar
[32]Yoshida, K., Nishimura, N. and Kobayashi, S., Application of new fast multipole boundary integral equation method to elastostatic crack problems, J. Struct. Eng., 47 A (2001), 169–179.Google Scholar
[33]Nishimura, N., Fast multipole accelerated boundary integral equation methods, Appl. Mech. Rev., 55 (2000), 299–324.Google Scholar
[34]Lai, Y. S. and Rodin, G. J., Fast boundary element method for three-dimensional solids containing many cracks, Eng. Anal. Bound. Elem., 27 (2003), 845–852.Google Scholar
[35]Wang, H., Lei, T., Li, J., Huang, J. F. and Yao, Z., A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations, Int. J. Numer. Meth. Eng., 70 (2007), 812–839.Google Scholar
[36]Liu, Y., Fast Multipole Boundary Element Method: Theory and Applications in Engineering, Cambridge University Press, New York, 2009.Google Scholar
[37]Peach, M. and Koehler, J. S., The forces exerted on dislocations and the stress fields produced by them, Phys. Rev., 80 (1950), 436–439.CrossRefGoogle Scholar
[38]Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J. M., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. and Vorst, H. van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Ed., SIAM, Philadelphia, 1994.Google Scholar
[39]Bettess, J. A., Economical solution technique for boundary integral matrices, Int. J. Numer. Meth. Eng., 19 (1983), 1073–1077.Google Scholar
[40]Saad, Y. and Schultz, M. H., GMRES: a generalized minimum residual algurithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869.Google Scholar
[41]Mullen, R. L. and Rencis, J. J., Iterative methods for solving boundary element equations, Comput. Struct., 25 (1987), 713–723.Google Scholar
[42]Phan-Thien, N. and Kim, S., Microstructure in Elastic Media: Principles and Computational Methods, Oxford University, New York, 1994.Google Scholar
[43]Valente, F. P. and Pina, H. L., Iterative techniques for 3-D boundary element method systems of equations, Eng. Anal. Bound. Elem., 25 (2001), 423–429.Google Scholar
[44]http://fastmultipole.org/.Google Scholar
[45]Huang, J., Jia, J. and Zhang, B., FMM-Yukawa: an adaptive fast multipole method for screened Coulomb interactions, Comput. Phys. Commun., 180 (2009), 2331–2338.Google Scholar
[46]Pimpinelli, A. and Villain, J., Physics of Crystal Growth, Cambridge University Press, New York, 1998.Google Scholar