An Efficient High Order Method for Dislocation Climb in Two Dimensions

We present an efficient high order method for dislocation dynamics simulation of vacancy-assisted dislocation climb in two dimensions. The method is based on a second kind integral equation (SKIE) formulation that represents the vacancy concentration via the sum of double layer potentials and point sources located at each dislocation center, where the climb velocity of each dislocation (or the strength of each point source) is proportional to the integral of the unknown density on the boundary of each dislocation. The method discretizes the interfaces only. Furthermore, it avoids the need of introducing additional unknowns as compared with the formulation in [11]; and unlike the formulation in [31] in which the kernel has logarithmic singularity, the boundary integrals involved in the formulation are easily discretized via the trapezoidal rule with spectralaccuracy. Thus, the number of unknowns in the linear system to achieve certain accuracy is optimal for typical settings in dislocation dynamics. We compare three different ways of solving the resulting linear system and demonstrate via numerical examples that the fast direct solver (FDS) in [20, 22] performs best for dislocation arrays, while the fast multipole method (FMM) accelerated iterative solver on the low accuracy FDS preconditioned system performs well for the general setting and exhibits roughly optimal complexity.