Adaptive Multiscale Coupling Methods of Molecular Mechanics based on a Unified Framework of a Posteriori Error Estimates

Multiscale coupling methods are significant methodologies for the modeling and simulation of materials with defects, intending to achieve the (quasi-)optimal balance of accuracy and efficiency. The a posteriori analysis and corresponding adaptive algorithms play a crucial role in the efficient implementation of multiscale coupling methods. This paper proposes a unified framework for residual-based a posteriori error estimates that can be applied to general consistent multiscale coupling methods. In particular, we prove that the error estimator based on the residual force can provide the upper bound of the true approximation error. As prototypical examples, we present a variety of adaptive computations based on this reliable error estimator for the blended atomistic-to-continuum (a/c) coupling methods, including the energy-based blended quasi-continuum (BQCE), the force-based blended quasi-continuum (BQCF) and the recently developed blended ghost force correction (BGFC) methods. We develop a coarse-grained technique for the efficient evaluation of the error estimator. A robust adaptive algorithm is therefore proposed and validated with different types of crystalline defects, some of which are not considered in previous related literature on the adaptive a/c coupling methods. The results demonstrate that the adaptive algorithm leads to the same optimal convergence rate of the error as the a priori error estimate, but with considerable computational efficiency. This study provides valuable insights into the design and implementation of adaptive multiscale methods, and represents a significant contribution to the literature on a/c coupling methods.