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.

翻译：多尺度耦合方法是缺陷材料建模与仿真的重要方法论，旨在实现精度与效率的（准）最优平衡。后验分析与相应自适应算法在多尺度耦合方法的高效实现中起着关键作用。本文提出了一个基于残差的统一后验误差估计框架，可应用于一般相容性多尺度耦合方法。特别地，我们证明了基于残余力的误差估计量能够提供真实近似误差的上界。作为典型示例，我们基于该可靠误差估计量，对原子-连续介质（a/c）混合耦合方法（包括基于能量的混合准连续（BQCE）、基于力的混合准连续（BQCF）以及新近发展的混合鬼影力矫正（BGFC）方法）开展了多种自适应计算。我们开发了用于高效评估该误差估计量的粗粒化技术，并据此提出了鲁棒的自适应算法。通过不同类型晶体缺陷（其中部分缺陷在以往自适应a/c耦合方法相关文献中未被考虑）的验证，结果表明该自适应算法实现了与先验误差估计相同的误差最优收敛率，同时显著提升了计算效率。本研究为自适应多尺度方法的设计与实现提供了重要见解，并对a/c耦合方法文献作出了重要贡献。