Efficient numerical methods for dislocation dynamics simulation

Abstract

Dislocations are line defects in crystals. They are primary carries of crystal plasticity. The direct simulation of dynamics and interactions of dislocations, known as dislocation dynamics, provides a promising tool for investigations of plastic behaviors of crystalline materials. However, for the method of dislocation dynamics to be a practical engineering tool, significant efforts are needed to improve the efficiency of dislocation dynamics methods. In this thesis, we have developed efficient numerical methods for dislocation dynamics simulations. First, we apply new version of fast multipole method to calculate stress field of dislocation ensembles in infinite mediums. Numerical experiments show that for a dislocation ensemble discretized into N dislocation segments, the new version of the FMM is asymptotically O(N) with an optimized prefactor, and very efficient for prescribed accuracy requirements. Second, we derive accurate approximation formulas to compute the contribution to the stress of a curved dislocation segment containing or very close to the target point within the framework of distribution of dislocations in small tubes. Previously in the literature, lots of numerical grid points for accurate calculation of such contribution. Finally, we develop an efficient numerical method that employs the fast multipole accelerated boundary integral equation method to solve the complementary boundary value problem of the elasticity system. The boundary integral equation method reduces the unknowns to those only on the material boundaries. Then the new version of fast multipole method is employed to compute convolution of the Green’s functions and the unknowns in a very efficient way