Periodic Boundary Conditions for Dislocation Dynamics Simulations in Three Dimensions
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Periodic Boundary Conditions for Dislocation Dynamics Simulations in Three Dimensions Vasily V. Bulatov1, Moon Rhee1, and Wei Cai2 1 Lawrence Livermore National Laboratory, University of California 2 Massachusetts Institute of Technology. ABSTRACT This article presents an implementation of periodic boundary conditions (PBC) for Dislocation Dynamics (DD) simulations in three dimensions (3D). We discuss fundamental aspects of PBC development, including preservation of translational invariance and line connectivity, the choice of initial configurations compatible with PBC and a consistent treatment of image stress. On the practical side, our approach reduces to manageable proportions the computational burden of updating the long-range elastic interactions among dislocation segments. The timing data confirms feasibility and practicality of PBC for large-scale DD simulations in 3D. INTRODUCTION Treatment of boundary conditions is an important element of Dislocation Dynamics (DD) methodology. There are two distinct classes of DD simulations that necessitate different approaches to boundary conditions. When the simulation volume is close to an internal or external interface (surface, crack, grain or phase boundary, etc.), it is necessary to account for stress variations associated with the interface. In other cases, dislocation behavior in the bulk single crystal, far removed from any interfaces, is of interest. The first case is generally difficult requiring the use of sophisticated numerical methods [1,2] to calculate the elastic (image) stress associated with the interfaces. In the second case, the material volume can be regarded as a small part of an infinitely large single crystal justifying the use of the relatively simple analytical solutions of the continuous theory of dislocations obtained for the infinite elastically homogeneous solid [3]. Understandably, in the early stages of development, DD simulations focused on the simpler case of bulk single crystals. Many years ago von Karman suggested a trick by which a small representative volume of material is replicated by periodic continuation to make up the infinite crystal and to preserve its translational invariance. Since then, this trick has been routinely employed in computer simulations of solids. In 2D, where dislocations appear as point objects carrying tensorial (Burgers) charges, periodic boundary conditions (PBC) have been successfully implemented [4,5]. However in 3D, the models used for DD simulations in the bulk remain inconsistent with the absence of material interfaces and translational invariance of the infinite crystal. In fact, most simulations performed so far employ free boundaries. To simplify the treatment, the boundaries are allowed to exist only in the sense that they absorb dislocations that happen to approach the boundary, whereas the image stresses induced by the free surfaces are simply ignored [6]. In order to mitigate such undesirable effects, a smaller spherical volume in the center of the simulation box is sometimes used to control th
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