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A Numerical Scheme for Generalized Peierls-Nabarro Model of Dislocations Based on the Fast Multipole Method and Iterative Grid Redistribution

Part of: Equilibrium (steady-state) problems Integral equations, integral transforms Generalities, axiomatics, foundations of continuum mechanics of solids Partial differential equations, boundary value problems

Published online by Cambridge University Press:  23 November 2015

Aiyu Zhu ,
Congming Jin ,
Degang Zhao ,
Yang Xiang  and
Jingfang Huang
Aiyu Zhu
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Congming Jin
Affiliation:
Department of Mathematics, Zhejiang Sci-Tech University, Zhejiang, China
Degang Zhao
Affiliation:
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
Yang Xiang*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Jingfang Huang
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA.
*
*Corresponding author. Email addresses:maayzhu@ust.hk (A. Zhu), jincm@lsec.cc.ac.cn (C. Jin), dgzhao@hust.edu.cn (D. Zhao), maxiang@ust.hk (Y. Xiang), huang@email.unc.edu (J. Huang)
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Abstract

Dislocations are line defects in crystalline materials. The Peierls-Nabarro models are hybrid models that incorporate atomic structure of dislocation core into continuum framework. In this paper, we present a numerical method for a generalized Peierls-Nabarro model for curved dislocations, based on the fast multipole method and the iterative grid redistribution. The fast multipole method enables the calculation of the long-range elastic interaction within operations that scale linearly with the total number of grid points. The iterative grid redistribution places more mesh nodes in the regions around the dislocations than in the rest of the domain, thus increases the accuracy and efficiency. This numerical scheme improves the available numerical methods in the literature in which the long-range elastic interactions are calculated directly from summations in the physical domains; and is more flexible to handle problems with general boundary conditions compared with the previous FFT based method which applies only under periodic boundary conditions. Numerical examples using this method on the core structures of dislocations in Al and Cu and in epitaxial thin films are presented.

Keywords

DislocationelasticityPeierls-Nabarro modeliterative grid redistributionfast multipole method

MSC classification

Secondary: 65R20: Integral equations 65N50: Mesh generation and refinement 74A50: Structured surfaces and interfaces, coexistent phases 74G65: Energy minimization
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1282 - 1312
Copyright
Copyright © Global-Science Press 2015 

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