It was pointed out that the use of periodic boundary conditions for modeling crystal dislocations was predicated on the ability to handle the inevitable image effects. The present work dealt with an often-overlooked mathematical subtlety which was involved in dealing with periodic dislocation arrays. This was the conditional convergence of the lattice sums of image fields. By analyzing the origin of conditional convergence and the numerical artefacts associated with it, a mathematically consistent and numerically efficient procedure was established for regularization of the lattice sums and the corresponding image fields. The regularized solutions were free from the artefacts caused by conditional convergence and regain periodicity and translational invariance of the periodic super-cells. Unlike the other existing methods, the present approach was applicable to general anisotropic elasticity and arbitrary dislocation arrangements. The capabilities of this general methodology were demonstrated by application to a variety of situations encountered in atomistic and continuum modeling of crystal dislocations. The applications include introduction of dislocations into the periodic super-cell for subsequent atomistic simulations, atomistic calculations of the core energies and the Peierls stress and continuum dislocation dynamics simulations in 3 dimensions.

Periodic Image Effects in Dislocation Modelling. W.Cai, J.Chang, J.Li, S.Yip, V.V.Bulatob: Philosophical Magazine, 2003, 83[5], 539-67