Order-Nmethod for force calculation in many-dislocation systems

We propose a methodology for calculating the elastic interaction forces of a large (N∼${10}^6$ or more) number of dislocations in a solid, for the purposes of molecular-dynamics type modeling of problems such as dislocation patterning and strain hardening. In order to avoid the ${\mathit{N}}^2$ scaling resulting from the 1/r decay of the forces, we introduce a variation of the particle-particle–particle-mesh method of Hockney and Eastwood [Computer Simulations Using Particles (Institute of Physics, Bristol, 1988)], applicable to arrangements of straight, parallel dislocations; extension to fully three-dimensional problems is discussed. Our version handles the anisotropic interdislocation forces with high accuracy and is easy to implement. Precision estimates are obtained; they indicate a very favorable scaling of CPU time with accuracy, which is supported by numerical results.