Adaptive Quasicontinuum Methods and Simulations for Crystal Defects with a Theory based Unified a Posteriori Error Estimate

Abstract: Adaptive quasicontinuum (QC) methods are important methodologies in molecular mechanics for the simulations of materials with defects, intending to achieve the optimal balance of accuracy and efficiency on the fly. In this study, we propose a residual-force based a posteriori error estimate that is simple and is unified for consistent quasicontinuum methods, as opposed to the widely adopted residual-stress based a posteriori error estimates which are complicated and need to be derived for the particular QC method under consideration. The simple and unified formulation of the estimator, together with certain sampling techniques, leads to a highly efficient and adaptable implementation. We also prove in theory that the unified error estimator provides an upper bound of the true error. We develop adaptive algorithms based on this unified estimator and validate the algorithms by several representative quasicontinuum methods for various types of crystalline defects, in terms of convergence and efficiency. In particular, the adaptive simulations of the anti-plane crack, of which we possess little a priori knowledge, demonstrate the necessity and significance of the proposed a posteriori estimates and the adaptive strategies.