On flexible Green function methods for atomistic/continuum coupling

Atomistic/continuum (A/C) coupling schemes have been developed during the past twenty years to overcome the vast computational cost of fully atomistic models, but have not yet reached full maturity to address many problems of practical interest. This work is therefore devoted to the development and analysis of flexible Green function methods for A/C coupling. Thereby, the Green function of the harmonic crystal is computed a priori and subsequently employed during the simulation of a fully nonlinear atomistic problem to update its boundary conditions on-the-fly, based on the motion of the atoms and without the need of an explicit numerical discretization of the bulk material.

The first part is devoted to the construction of a discrete boundary element method (DBEM) which bears several advantages over its continuous analog, i.a. nonsingular Green kernels and direct application to nonlocal elasticity. As is well-known from integral problems, the DBEM leads to dense system matrices which become quickly unfeasible due to their quadratic complexity. To overcome this computational burden, an implicit approximate representation using hierarchical matrices is proposed which have proven their efficiency in the context of boundary integral equations while preserving overall accuracy. In order to solve the coupled atomistic/DBEM problem, several staggered and monolithic solution procedures are assessed. An improvement of the overall accuracy by several orders of magnitude is found in comparison with naive clamped boundary conditions.

To further account for plasticity in the continuum domain the coupled atomistic/discrete dislocations (CADD) method is examined, including the treatment of hybrid dislocation lines that span between the two domains. In particular, a detailed derivation of a quasi-static problem formulation is covered and a general algorithm to simulate the motion of the hybrid dislocations along A/C interfaces is presented. In addition, to avoid solving the complementary elasticity problem, a simplified solution procedure, which updates the boundary conditions based on the Green function of the entire dislocation network for obtaining accurate stress and displacement fields, is introduced and validated. The test problem consists of the bowout of a single dislocation in a semi-periodic box under an applied shear stress, and excellent results are obtained in comparison to fully-atomistic solutions of the same problem.